To set the stage:

First, we can broadly divide the history of al-Andalus into several stages: (1) rough unity under the Umayyads (first as a branch of the Damascus caliphate; then independently as the "caliphate of Cordoba") (2) the 11th century "fitna" or a period of divided rule by so-called taifa kingdomgs (3) rule by the North African-based Almoravid and Almohad dynasties, punctuated by another fitna (4) just a few independent Muslim-ruled holdouts as Christian Spain finalized its Reconquista.

Second, at no time did Muslims rule the entire Iberian peninsula. Thomas Glick observes that the dividing line was roughly climatological as well, with al-Andalus occupying the more Mediterranean, olive-friendly climate zones and Christian huddled up in the cereal-growing, Atlantic Isles-y north.

By way of "international relations":

Sources are always going to be a problem for the earlier period. Starting in the 9th and especially 10th centuries, we have a clear picture of Umayyad al-Andalus woven into a circum-Mediterranean Islamic trade zone. There are some signs of maritime trade between Andalusi ports, Venice, and Constantinople. More so than Muslim ships going to Christian ports were Christian and Jewish merchants coming to southern Iberia. Its overland trade with Christian Europe seems to have been almost entirely with the numerous Christian kingdoms or principalities in its north.

The border principalities paid a tribute tax of sorts to the Umayyads. By way of trade, Olivia Remie Constable sketched out a relationship you might recognize: Christian Iberia sent commodities to the south, al-Andalus; luxury goods and more manufactured items went north along with, eventually, gold from the West African trade.

During the more anarchic fitna era, Islamic and Christian kingdoms alike warred over the Iberian frontier, making and breaking alliances with each other. Tribute payments also went both ways dependent on power.

The Almoravid and Almohad dynasties restored a semblance of order to the geographically-shrinking Islamic Iberia. The revival of Latin Europe's economy especially after 1000 (with some signs beforehand, of course) reoriented the European economy. Rising powers Genoa and Pisa became key trading partners for the 11th-13th century Muslim Iberians. Interestingly, it seems to be the Italians who sought to win/buy trading privileges with al-Andalus, not the other way around.

The later dynasties also hired Christian mercenaries and mercenary groups for their armies--a marked shift from how these two groups treated their native Christian populations...

By way of internal interreligious relations:

Islamic law starting with the Quran famously establishes Christians and Jews as dhimmis or peoples of the book: subject to restrictions on religious displays and an extra tax (jizya) BUT tolerated and allowed to continue religious practices. So much for the ideal. What did it look like on the ground?

Under the Umayyads, sources hint at an atmosphere of relative toleration and only a gradual attrition of Christians converting to Islam. Initially, in fact, the Umayyads discouraged the lower Christian aristocracy from converting. They weren't powerful enough to pose a threat or really affect the course of al-Andalus, and the increased tax was quite nice for Andalusi coffers. Certainly the most powerful Christians lost their standing. Andalusi courts were centers of intellectual life and culture that brought together primarily Muslims, but also Christiand Jewish scholars. I've read that intermarriage between Christians and Muslims was fairly common, although I think this would have been primarily or exclusively Muslim men and Christian women (I am extrapolating from later medieval sources here based on continuity of law, so I am VERY open to being corrected!) Over the centuries, many "moriscos Mozarabs" or Andalusi Christians did end up converting to Islam, and why not? Scholars have talked about "conversion by the coin" instead of by the sword or by faith, and that reflects the picture we have from elsewhere in the Mediterranean.

The Almoravids and especially the Almohads were different. By the 11th and especially 12th-13th centuries, large swathes of the Mediterranean world seem to have been focused more on uniformity of belief and exclusion of alternate perspectives. This applied internally as well as externally. Even as they invited Christian mercenaries from abroad to pad the ranks of their armies, the Almoravids clamped down on Muslims perceived as less-than-pious. And it is sources from the Almohads themselves that describe their brand of religious zealotry as "conversion or death"! This would, again, apply to less-pious Muslims, but tragically to Jews and possibly Christians as well. We know of several massacres against Jews who refused conversion.

As Christian armies gnawed away at the last vesetiges of Islamic Iberia in the 13th-15th centuries, they would have found very little native Christian population to greet and/or fear them. By that point, Christians (and many Jews) had already fled north or converted.

QUESTION 1

Ok, so let's address the first question first - you've essentially given an extremely broad area that can lead to very different responses depending on the era, so I'm going to just pick one to focus on. In particular, I'm going to look at the relations with the various Christian groups to the north during the reign of Abdul Rahman III, the first Cordoban Caliph (912-960). This era in particular is very important in the history of al-Andalus (that's what the Andalusis called it!) and Abdul Rahman III is central to the entire Caliphate of Cordoba era. It also helps that his reign in particular is a specialty of mine, so there's also that (I can talk about him all day if you give me the chance).

Here I'll refer to the frontier regions which refers to the northern border of al-Andalus which saw the Navarrans, Leonese, and Castilians in particular on the other side. The first decade or two of Abdul Rahman III's reign was spent mainly dealing with various rebellions which are very interesting but I won't get into them here since it's off topic. So a lot of time was spent consolidating and defending, building up to return to stability that would allow the Caliphate to flourish incredibly later on. Maribel Fierro (in the book "Abd Al-Rahman III: The First Cordoban Caliph" - an excellent intro to al-Andalus and the Caliphate in particular if you're interested!) also rightly refers to his aim in the first three decades as being to defend and maintain the frontiers. Basically, at that time there was no interest whatsoever in expanding the territory under his rule because there had been so many troubles before with rebellions here and there, and Abdul Rahman III himself very nearly avoided death in an ambush that occurred in battle with rebels at one point so he simply at that point did not care to expand the borders.

There was conflict of course, but in the first thirty years or so Andalusi roles in the frontier regions were pretty much always responses to the various activities of the Navarrans, Leonese and Castilians. What I mean by this is that Andalusi campaigns at this point usually only happened after one of the northern polities tried to expand their borders or launched an attack. They did this very opportunistically, of course. Usually they waited until the Umayyad armies were occupied fighting rebels down south in the central areas of al-Andalus. They also took heavy advantage of the fact that the frontier lords were often, as was common in western europe at the time, often engaged in internal conflicts against one another. The frontier lords of al-Andalus were willing to ally with the Christians not only against each other but also against attempts from Cordoba to centralize and limit the autonomy of the frontier lords. So tangled webs of opportunistic alliances along the frontiers were very common, and during this time they weren't discriminating at all. Let's take the example of Toledo. In some of the period sources from the early years of Abdul Rahman III's reign we often see them writing about how the people in Toledo were always willing and eager to involve Christians in their own affairs, and any time they had a problem with the Caliph they would run to the christians north to get their aid (which was inevitably provided).

In fact during this period when the frontiers shifted in favour of the northern christian kingdoms, it was usually because of (a) tensions between Cordoban rule and the frontier lords who wanted autonomy and (b) Game of Thrones style powerplays and plots between all the frontier noble families and lineages. So the frontier lords often didn't care about their king or what he wanted if it meant their autonomy would be reduced to an extent they couldn't handle. So the answer to the question of, were there alliances? You bet your ass there were - in fact what we generally don't find in this era are frontier lords who didn't engage in alliances with the Christian polities.

What about diplomatic relationships? Did they engage in those too? Well, obviously all the military activity and alliances I just discussed goes hand in hand with diplomatic relationships. Treaties and alliances basically were being signed all the time with Christian kingdoms for all sorts of reasons. Take the example of the Caliph's campaign in 934 against northern rebels who were allied with some local northern christian lords. The Navarran Queen Toda contacted Abdul Rahman III seeking an alliance and hoping for the Caliph's approval and endorsement of her son's (Garcia Sanchez( inheritance to the Navarran throne. The Caliph agreed, and subsequently acknowledged Garcia Sanchez as the king of Navarra. Later on in 958 Queen Toda and Abdul Rahman III had further relations when her grandson, the king of Leon (Sancho the Fat) was overthrown by the nobility and went to her for refuge. They both travelled to Cordoba and gave their respect and submission followed by a treaty being signed, whereupon I've also read that sources describe him as having been cured of his obesity by a prominent Jewish doctor named Hasday ibn Shaprut who was also a high ranking official in the Caliphate. Consequently with the Caliph's backing in 960 he retook the throne, although eventually he broke the treaty and his reign was thus marked by even further rebellion from nobility who were angered by it.

You might be interested to know that relations like this would often occur because Andalusis and non-Andalusi Iberians often had common heritage due to centuries of intermarriage. Abdul Rahman III himself had Basque heritage, his mother having been a Christian concubine originally from the north. It was on this basis that Queen Toda of Navarra claimed common heritage with Abdul Rahman III, as his grandmother was the daughter of Fortun Garces. Many people on both sides recognised this fact and it often played role in diplomacy when there was demonstrable commonality in lineage.

What other examples can I give you? Well, take the treaty signed between Abdul Rahman III and the king of Leon in 935 that ensured the people in Zaragoza could not rely on Leonese help in conflict with the Caliph. The Jewish doctor I mentioned before, Hasday ibn Shaprut, was one of the Caliph's preferred diplomats who negotiated extensively with Christians in signing treaties. Besides peace treaties, extensive diplomacy was conducted by Hasday ibn Shaprut on the Caliph's behalf to free individuals from imprisonment (such as in 940 when he spent 7 months in the Leonese capital to get prisoners freed and sign a peace treaty). In the same year, he went to Barcelona to sign a treaty guaranteeing trade and commerce between the local counts as well as a diverse array of southern French counts who also wished to trade extensively with the Andalusis.

But let's not forget in your question that you said "surrounding Christian powers", so I'm going to take the opportunity to extend my answer to powers outside Iberia, but I'll try to keep it "short" (by the standards of an AskHistorians post). Anyway, so there were embassies sent to Cordoba quite a few times. A notable one was an embassy from Sardinia, where the Sardinians sought a formal peace treaty with the Caliphate, mainly a symbolic thing so they could expand trade a lot. This embassy was accompanied by a large group of Italian merchants from Amalfi who initiated trade & commercial operations. Frankish, German, and Eastern Roman ambassadors and embassies were also notable arrivals in Cordoba. Abdul Rahman III actually sought an alliance with the Romans in order to launch a comprehensive multi-front war to fight off the Fatimids, although this alliance did not happen as Constantine VII Porphyrogenitus (Emperor at the time) decided to honour the armistice with the Fatimids as he did not wish to risk losing Roman holdings in the Italian peninsula. One thing I always thought was cool was Constantine VII sent some rare and very valuable books to Abdul Rahman III as gifts, one of which was a pharmocology book written in Greek by Discorides. The Caliph responded by asking for a translator as he didn't know Greek, and couldn't find any Andalusis who did either so eventually a Christian monk named Nicolas arrived and worked with, guess who? Yes, you guessed it - our main man, Hasday ibn Shaprut. Gotta love that guy.

Furthermore, we also find that there were German and Frankish embassies sent, and there's a pretty famous account of the Frankish ambassadors marvelling in awe at the wealth and beauty of the palace grounds of Madinat al-Zahra. Although these accounts are somewhat embellished as people passed around stories and praised Abdul Rahman III, of course. King Otto I from Germany had sent several ambassadors to conclude treaties with the Andalusis, although the first was made to wait three years before getting to see Abdul Rahman III as King Otto's first letter was seen as being insulting to the Prophet. Otto eventually sent Recemund, a Christian who Abdul Rahman III liked so much that he was eventually appointed as the bishop of Elvira. Recemund later served as a diplomat for the Caliph and travelled to the Eastern Roman Empire, among other places.

So basically, the notion that there was some sort of line drawn in the sand of "christians vs muslims" who didn't interact outside of war with each other is demonstrably false. Such a construction of politics didn't really occur until the crusades began. During the Umayyad era it hardly existed if at all, and relations on the frontier were pretty standard for the early middle ages regardless of religion. Relations were complex and rarely based on religion. However, I'd like to reiterate that after the Cordoba Caliphate and the fitna of andalus, things began to change as we enter the later parts of the middle ages. The Almoravids and Almohads existed in a different world, and religion was more important -that's a bit outside the realm of my expertise though.

within the Caliphate?

to clarify are you asking about the caliphate specifically or the broader history of islamic-Christian political relationships?

You are basically asking about relationships between one to dozens Muslim polities and up to maybe a dozen distinct Christian polities over the course of almost eight centuries (720s to 1492). Is there any particular period you are interested in?

Also, in the second question, do you mean the Cordoba caliphate in Al-Andalus (proclaimed in 929)? Or the Caliphate that was centered in the Middle East?