I'll be focusing on your first question, as the second question is connected with a very broad time-frame of more than 700 hundred years of Islamic realms in the Iberian peninsula. The short answer regarding if Muslims could continue to live in Spain after the reconquista would be: Yes, if they converted to Christianity. But (as always) it's more complicated than that.
Let's first look at the capitulation of Granada in 1491, the last Islamic Iberian realm. The capitulation was set down by “the Catholic Monarchs” Ferdinand II of Aragon and Isabella of Castile. It built on earlier precedents and was relatively favorable: The conquered people received the status of mudéjares, and as common in this case their religious and cultural traditions were confirmed. This included the continued application of Islamic sharia laws, but also a general amnesty for all crimes committed during the conquest wars. For the most part traditional elites should stay at the top of their communities and properties should be left untouched. Those Muslims who wanted to leave to “Barbary” (Northern Africa) were allowed safe passage during the next three years, and they could still emigrate after that against payment – leading to large emigrations. In general, Muslims were not supposed to be converted to Christianity against their will (in stark contrast to the expulsion of all Iberian Jews in 1492).
This relatively tolerant capitulation can be seen in connection with a medieval Iberian mentality, positing that Christian and Muslim states had to coexist next to each other. By now Castile did not have to fear campaigns by Granada anymore, and the capitulation thusly only applied for a short time. In the late 15th century Christian immigration to Granada was facilitated by the Crown, and the capital was divided into a Christian and a Muslim part in 1498. The turning point came with the increasing influence of archbishop Francisco Jiménez de Cisneros. Cisneros' measures including forcibly „bringing back“ converted Christians to Christianity; forced conversions of the Islamic elite; the burning of arabic, esp. Religious books. These measured led to an uprising at the Albaicín, which in turn led to mass conversions of 50-60.000 people – marking the beginning of crypto-Islam in Iberia. The resistance in Granada influenced the falied rebellions in the Alpujarras from 1500, which could be used by Christian authorities as an argument to abolish the capitulation and to change or abolish the mudéjar status. The whole population of Granada and consequently of Castile was left with the „choice“ to either emigrate or convert. With the end of official Islam in Iberia the converted Muslims were referred to as moriscos or cristianos novos („new Christians“).
The former mudéjares were under the inquisition's jurisdiction and had little possibilities for action – either accepting Christianity, emigrating (mostly illegaly) or continuing to practice Islam clandestinely. Calls for help to other Muslim realms like the Ottomans were to no avail. Teachings of Christianity and its traditions did not prove to be effective, e.g. due to language barriers and lack of priests. While in the first half of the 16th c. there was still a policy of relative tolerance, exclusions from offices and proffessions proliferated. Later policies were geared towards assimilation through suppression of customs rather than of religious pracices.
The second important turning point came with the accession to the Spanish throne of Philipp II. In 1556. It signaled the end possibilities for moriscos to buy freedom from the inquisition and from repression. More aggressive crown policies looked towards open repression instead of christianisation – by the synod of 1565 and later reforms the goal was to eliminate all signs of Muslim religion and culture. The 'Pragmática Sanción' is of special importance here, which left the moricos 12 months to adopt Christian mores. This change in official policy and the failure of diplomatic missions by the moriscos left the latter with few options indeed. The 2nd revolt of Alpujarras starting in 1568 has to be seen before this background. This revolt turned into a full-fledged war until 1571 spanning nearly the whole kingdom of Granada. To put it shortly, this war necessitated Philipp's intervention, who sent his brother Don Juan de Austria with ca. 20.000 man to quell the uprising, declaring campo franco.
The 2nd revolt made a return to coexistence improbable if not impossible. In a first step it led to large-scale resettlements of Granada's moriscos to other regions. The illegal emigrations understandably increased in this time. While ca. 165.000 moriscos had lived in Granada before the revolt, around 80.000 of them were resettled, while the rest left or died during the conflicts (the numbers are notoriously hard to determine). The morisco population continued to decrease in the following decades. In a second step came the final expulsion of all moriscos from Spain from 1609 onwards. This inclueded around 120.000 people from Valencia who were expulsed to Northern Africa, and in lesser numbers from Aragon and other regions. As many moriscos esp. in Castile resisted the operations took until 1614. The majority left for Morocco, Oran, Algier and Tunis, and others emigrated to Thessaloniki, Istanbul and the Balcan. Although moriscos continued to return clandesinely in smaller numbers, the expulsions of 1609 are usually considered the end-point for crypto-Islam in Spain.
Short version: There were different phases regarding the Muslim population's treatment. Following Granada's capitulation in 1491 the continued exertion of Islamic pracices and customs was allowed. This changed at the turn of the century with mass conversions, leading to continuing illegal immigration and clandestine practice of Islam by the converted “moriscos”. The last phase coincides with the 2nd revolt of Alpujarras from 1568 which led to stronger repression. The moriscos had little to no political leverage in Spain, and were expulsed by the Spanish Crown in 1609-14. The changes in policy were connected with a general sharpening of religious differences, esp. with the beginning of Philipp II's reign.
Sources:
Del Marmol Carvajal, Luis, Historia del rebelion y castigo de los Moriscos del reino de Granada, Madrid 1797.
De Epalza, Mikel, Los moriscos antes y despues de la expulsion, Madrid 1992.
Caro Baroja, Julio, Ciclos y temas de la Historia de Espana: Los moriscos del Reino de Granada: Ensayo de Historia Social, Madrid 1985.
Echevarria, Ana, The Attitude Towards Muslims in Fifteenth Century Spain, Leiden 1999.
Harvey, Leon Patrick, Islamic Spain: 1250 to 1500, Chicago 1990 & Muslims in Spain: 1500 to 1614, Chicago 2005.
Nueva Recopilacion de Leyes de Castilla de 1567, Edicion Facsimilar del Boletin Oficial del Estado.
Now that /u/drylaw covered the latter part of the story - something that I frankly dreaded to do as it's so far outside of the period that I study - let me have a closer look at what happened before the late 15th or, God forbid, the 16th century.
Let's start with carefully rephrasing your question so that it is easy to answer. How likely was it for a Spanish Muslim to be allowed to stay and live peacefully after the Christian (re)conquest of their town or village? Oh, absolutely. Muslim areas were reconquered over a period of at least four centuries (leaving aside the early 9th century Carolingian conquest of what is now Catalonia where we simply don't know much about what actually happened). Once the process was under way, roughly starting with the collapse of the Caliphate in the early 11th century, surrendering Muslim towns agreed to terms of capitulation wherein their new Christian overlords promised to basically leave the Muslims be. (They didn't actually had a choice as there was just much more Muslims than Christians, particularly early on.)
Note that at least until the 12th century these towns by the time of (re)conquest would also have Jewish and Christian communities, so not everybody was Muslim. But by the 13th century, when the bulk of the Reconquest happened, Muslims were the majority. Thus important Mudéjar (=Muslims living under Christian rule) communities were formed all over the peninsula. They kept their laws and judges (Christian judges only had jurisdiction in cases that involved both Muslims and Christians or Jews), and lived in their own neighbourhoods, known as aljamas or morerías, the same term applying both to their communities and the physical spaces that they occupied in Iberian towns. In some parts of Spain these communities actually remained the majority of the population even centuries after reconquest (mostly in Aragon and particularly in Valencia that at some point was considered to be home to almost half of all the Mudéjars in Spain). They were skilled craftsmen and very much in demand as builders and decorators. Just to give you an idea, [this beauty] (https://upload.wikimedia.org/wikipedia/commons/3/36/Fachada_de_la_Parroquieta.jpg) is not a mosque, as one would imagine. It is the side wall of a funerary chapel built at the cathedral of Zaragoza in Aragon for the powerful Archbishop Lope Fernández de Luna. The wall was [decorated] (http://gozarte.net/wp-content/uploads/2013/12/parroquieta-ventana.jpg) - redecorated, actually, as the Archbishop was not satisfied with the first wall that he thought was too plain - in 1378. That is, a full 260 years after the Christian conquest of Zaragoza (1118) and 130 years after the conquest of Seville (1248), the city that the two masters that did the wall came from. These communities survived, slowly eroding by opportunistic conversion, until the forced conversion that drylaw outlined above.
The real question is, however, were these Spanish Muslims of Arabic or Berber origin, as your question implies. And here it gets complicated. Really complicated. As in, we don't really know. But probably not really.
First, an important caveat. Our knowledge of the first centuries of Muslim domination of Spain and particularly of the actual conquest is really sketchy. Most information we have comes from Arabic sources that were written several centuries after the fact. (This, for example, allowed Richard Collins to largely dismiss them for his The Arab Conquest of Spain, 710-797 (Oxford 1989), although it should be pointed out that he doesn't read Arabic anyway).
That said, historians tend to think the majority of the initial invading force in 711 was composed of Berbers and other North African groups (quite possibly including some descendants of the Vandals who had gone through Spain three centuries before to found a kingdom in Africa). At least some of the officers and commanders were Arabs, though.
We know they came with their wives and families so they didn't really have to intermarry into the local population. We also know there were several instances of migration after the conquest, including a large wave straight from Syria whose arrival led to a civil war in 741.
We know that these Arab clans formed two larger opposing tribal groupings, referred to as 'Yemenis' and 'Kaisites', and that they settled in the valleys of the Ebro, Guadalquivir, around Toledo and in Murcia, grabbing the best land and booting the poor Berbers out to thinner pastures high in the mountains and to some God-forsaken places like Galicia (where said Berbers duly mutinied after a catastrophic draught, abandoning Galicia en masse and thus providing the nascent Christian Asturian kingdom with its first territorial gains). The Berbers were quite heavy in the interior of the Spanish Levant, i.e. in what is now the Valencian Community, too.
Even though it is frankly impossible to estimate how many Arabs and Berbers actually came to Spain, attempts have been made (somewhere between 30,000 and 50,000). Anyway it was a drop in the bucket, given that the entire population of Al Andalus may have easily numbered several millions (Glieck). As an aside, not all the Muslim newcomers in these first centuries were Arab or North African, either. A lot of people in the armies and, later, the upper administrative positions in Al Andalus were Saqaliba, i.e. slaves or emancipated eunuchs of Slavic and/or Western European origin.
With all these uncertainties, the traditional view has long been that Arabs and Berbers soon became a minority even among the Muslim population that in itself only became the majority in the Muslim-controlled lands by as late as the 12th century. The bulk of the Muslims would be of local Iberian stock, gradually converting to Islam over several centuries. Logically, one could assume these few Arabs and Berbers (and, later, some black Africans who may have come with the Almoravids and the Almohads in the 12th century) were quickly assimilated into a largely Iberian population, losing their Arabic or Berber identity.
Now this view has started being challenged some 40 years ago with emerging evidence that Arabs and Berbers did not actually just disappear into this largely Iberian mass. Part of this evidence was known all along but we chose to ignore it - like the fact that the Muslim army was organised along tribal lines well into the 10th century or that a period of civil war, the fitna, in the late 9th - early 10th century, was fuelled by the resentment that the Christians, Berbers, and, most of all, the ever growing Muwallads (=Iberian converts to Islam) felt towards the entrenched privilege of the Arab clans. Some evidence was quite new as historians like Pierre Guichard [1] actually brought in knowledge from adjoining fields like anthropology to point out the lingering effects of Arab and Berber social structures in medieval Al Andalus. Say, in the Levant a lot of place names start (or started at the moment of Christian reconquest) in Beni-, attesting to a separate settlement of a patrilinear Arab or Berber clan (Banu Something), etc.
All in all, until a new study comes that will absolutely change our views on this (and I would not be surprised if it happens soon), the consensus still seems to be that by the time a Christian army rode up to another Muslim stronghold to siege it or negotiate surrender the local Muslim population would be largely, hm, Muslim Spanish and not particularly Arab or Berber. But now we are somewhat less categoric about it than we used to be.
And now, for dessert, some fancy genetic stuff. A resent study of Spaniards and the Portuguese [2] shows that the Straight of Gibraltar did constitute an important barrier in gene flow. Yet about 10 per cent of the surveyed sample in the peninsula shows admixture from an 'African-type' population (compared to the samples that were taken in Tunisia and Morocco). Now these would be your Berbers and other North African groups. At the same time admixture from what they think is Sephardi Jewish type is twice as high, standing at 19 per cent. I must admit that I know next to nothing about genetics, but I wonder how they know this is a Sephardi Jewish signature and there's no Arab signal in there somewhere? From what we know about Spanish history, there would be.
[1] Guichard's pioneering research is summed up in his Structures sociales «orientales» et «occidentales» dans l’Espagne musulmane, Paris, 1977. This book is a gem.
[2] Adams, Susan M. et al. The Genetic Legacy of Religious Diversity and Intolerance: Paternal Lineages of Christians, Jews, and Muslims in the Iberian Peninsula, in: The American Journal of Human Genetics , Volume 83, Issue 6, pp. 725-736. Online at: http://www.cell.com/ajhg/abstract/S0002-9297(08)00592-2.
