This is a really interesting question, but it's pretty tough to answer. All these ethnic terms are really a vocabulary used by historians who study the past, rather than words that people in the medieval past would have used to describe themselves. And when they did use these kinds of words, it's often hard to figure out what they meant. For example, "Scotus" seems to have been used as a nickname for anyone who had been raised in Ireland, "Saxonus" could be used to describe anyone from England or from Saxony in Germany, and "Danus" meant anyone who spoke a variant of Old Norse, no matter where they were from (including the "Danes" of Irish descent who joined the Scandinavian immigrants who settled Dublin). Making things even more difficult from a historian's perspective, the texts that survive were almost all written by nobles transacting business or highly-literate clerics, so we don't know how the other 98% of the population might have used these terms.

In contrast, the way that we use these terms today was really defined in the 1800s, as governments became more complex and began to regulate their societies and cultures through things that we now take for granted, like public education. One of the major roles of early public education was teaching people that they all actually belonged to a single community, and that's really where nationalism took off. This is, for example, why many Americans still grow up reciting the Pledge of Allegiance in school. In Europe, nationalism was similarly defined through a shared sense of community and above all through history. Great Britain came from the Anglo-Saxons, France from the Franks, Spain from the Visigoths, Italy from the Ostrogoths, and so on and so forth. Significantly, almost all of these national histories began after the fall of Rome, so Britain or France or whomever could see themselves as the legitimate successors to Roman rule. The Roman Empire, as we know, was huge, and seeing yourself as a successor to Rome could help justify more modern acts of colonialism.

In short, these ethnic terms are a bit of a mess, not really appropriate for the middle ages, since if they were used, they probably meant something very different from how we use those terms today, whereas the way that we continue to think about these kinds of ethnic terms remain very closely entangled with 19th- and early 20th-century styles of nationalism and colonialism.

The term Celtic is especially interesting to me. The Celts were originally defined (at least for modern purposes) by the Romans, who were trying to make sense of the mess of peoples living on their northern frontiers. Celtic culture, like all cultures, was rapidly changing and not at all homogenous, but generally speaking the "Celts" spoke a different set of languages and lived in ways distinct from their Mediterranean neighbors. They probably didn't see themselves as much of a group, at least not until the Roman Empire began expanding, importing Mediterranean ways of seeing the world and going about day-to-day living (such as eating cilantro and fish sauce). Only then did they start to unify around the idea of being "Celtic" as an alternative to Roman rule. But the Roman forces prevailed in Gaul and carried Roman rule into Britain, while later missionaries brought the Christian legacy of Rome even into distant Ireland. But by this point, over 400 years after Julius Caesar, nobody was referring to the Celts any more. As I mentioned above, "Scotus" became (confusingly, at least for me) the preferred way for referring to anyone from Ireland.

Fast forward to the 1800s. The Potato Famine crushed Ireland, and the British overlords were completely ineffective at providing support. Many Irish fled to distant colonies, where they found themselves unwelcome. The Irish at home meanwhile continued to see themselves as an oppressed and colonized people. But as the British began to take pride in their heroic Anglo-Saxon past, some Irish realized that Ireland had had a lot going on at the same time. Exploring the medieval Irish past became a way to reject British dominance at home and retain pride among immigrant communities abroad.

Medieval Irish monasteries had churned out incredible works of art, exciting chronicles of their time, some of the most far-out Latin poetry ever recorded, and the missionaries who Christianized England and Germany. If you wanted to claim a glorious moment in the past as the moment when your people was born, medieval Ireland was a good place to start. Four masterpieces in particular became the foci for Irish pride: the Book of Kells, the Ardagh Chalice, the Cross of Cong, and the Tara Brooch.

Chances are, when you think of things that are Celtic, you think of things inspired by these four artifacts. The round Celtic script used at every Irish pub outside of Ireland draws its inspiration from the Book of Kells. The weaving loops of Celtic and tribal tattoos echo the finely wrought interlace of the Ardagh Chalice. The Catholic side of the Irish diaspora (and thus anti-Protestant and anti-British) carried replicas of the Cross of Cong to churches across the globe. And replicas of the Tara Brooch were used as insignia for Gaelic groups and independence fighters into the 20th century, and it became a stock image stitched into almost every representation of medieval heroes like Brian Boru.

But none of these things, unfortunately, is actually Celtic. These things are all inspired by medieval relics from Ireland, not the pre-Roman, pre-Christian tribes who were actually called Celts. Moreover, the culture that manufactured these relics—that is, the early medieval religious elite—were a remarkably international culture. The metal artifacts were all made of or included imported raw materials, whereas archaeologists now think the Book of Kells might actually have been made at a previously unknown monastery at Portmahomack in eastern Scotland. But that's not to downplay the importance of Celtic culture today, which is a modern reinvention of Irish culture based on a set of beautiful and awe-inspiring works of art. "Celtic" culture is by no means made up out of thin air. I only hope to show that modern manifestations of the "Celtic" past are actually an eclectic draw on a much more complex and exciting past than is commonly assumed.

So where does that leave in terms of your original question? Well, we should probably ditch the word Celtic if we're making comparisons with medieval peoples like the Anglo-Saxons or the Welsh. But the things that we think of as "Celtic" actually came from an international group of churchmen who absolutely would have been closely tied into the ecclesiastical communities across Scotland, England, and Wales, as well as into the heart of the Carolingian Empire and out into its distant frontiers in Germany. There do in fact seem to have been Irish and Roman factions in the early English church, although these were largely reconciled at the Synod of Whitby in 664. The languages are a bit more complex, with Irish Gaelic being related to what was then being spoken in Wales and Scotland, but probably not England, where beginning around the 500s they spoke a Germanic form of Old English very different from modern English which incorporates a lot of Latin elements borrowed from medieval French. Music is a big question mark, since we have only the scantest clues for music before the 1200s, when the earliest forms of modern music notation were first invented. All in all, I must admit that this is a messy answer, but I hope you can see why any simpler answer would necessarily be misleading. Thanks for reading!

Can't tell you about the culture, but I can tell you a little about the languages.

The Gaelic languages are a branch of Insular Celtic languages, and include Irish, Manx, and Scottish Gaelic. Gaelic was brought to Scotland from Ireland at about the same time as Old English. Before that, the inhabitants of Scotland spoke Pictish. Now, there have been conflicting theories about Pictish, but afaik the scholarly consensus now is that it was a Celtic language, but not a Gaelic nor a Brittonic one, being somewhat more distantly related to them. Since it has gone extinct quite a long time ago, and has left few traces, it's difficult to judge its exact position within the family with any precision.

Since both Scottish Gaelic and Scots have been spoken in Scotland for approximately the same time (one in the highlands and the other in the lowlands), that would likely be the reason why you heard of Scotland being "more Anglo-Saxon". The reality of it is, the linguistic history of the British isles is rather complex, and you can't just pick one single language that was the "main" language of Scotland.

tl;dr: Irish Gaelic, Scottish Gaelic, and Welsh are all Insular Celtic languages with demonstrable family ties to each other.

If you want to know more about these languages in particular, you can also try /r/linguistics, we have a few people who specialize in Gaelic linguistics.

The Irish are not Anglo-Saxon. There is no evidence for this so I don't know where you could have read this. Ireland was originally settled by migrant nomadic people who came into Ireland sometime after 10,000 BC and were followed by people with farming skills who arrived around 4,500 BC. There was a slow tricking in of people over the following centuries and the population built up. Neolithic sites such as Newgrange built around 3200 BC are evidence of the presence of a sophisticated people with knowledge of astronomy.

The so called 'Celtic' period is Iron Age and there is no evidence of a migration of people into Ireland during this time. What there is archeological and other evidence for is that the Celtic culture arrived, including the language, laws and religion but no migration of a 'Celtic' people.

The mythology of a Celtic people arriving in Ireland is mostly based on an ancient text called the Leabhar Gabhála Érenn which was taken by 19th century authors as history and the notion took hold. But it's not history, it's mythology and no Irish scholars now accept this document as history. But what is accepted is the presence of a Celtic culture in Ireland.

By the time that Christianity arrived in the fifth century and the historic period began, Ireland was a Celtic/Gaeilge speaking country. Texts written shortly after this attest to this. Written laws known as Brehon law attest to this also. Early texts of epic tales and poetry reveal a Celtic culture similar - but not identical- to that found in continental Europe

The Irish migrated into parts of Scotland in the sixth century and settled in the western area which became known as Dál Riata. With this settlement they brought the Irish language to Scotland and it became Scots Gaelic. The word Scotland originally meant 'land of the Irish' as Scotti was the Latin name for the Irish. The future kings of Scotland were descended from these Irish settlers.

There is no real evidence of a large-scale invasion into Ireland by Celtic tribes. What is more likely is that small groups of Celts or 'Celtified' people, arriving from Britain, Gaul and possibly also Iberia, imposed themselves on the local population, who were derived of a mixture of the Neolithic population and Bronze Age incursions of an Indo-European nature (horses, wheels, metalworking). It seems there were waves of these 'Celts', many probably pressured by Roman incursions into their territory and/or migrations by other tribes being pushed by the Romans.

The 'Celts' took charge, and the Gaelic language became a prestige language (similar to English in 19th-century Ireland) that, over time, became the dominant language on the island. But the language, religious practices, burial rites and other cultural elements showed remarkable continuity with a previous, pre-Celtic age.

The Origins of the Irish by Mallory is a good read on this topic, as is Pagan Celtic Ireland by Raftery.

The Breton, Welsh, Cornish and Gaelic languages are all Celtic.
That's unambiguous and not tough to answer.

You mention lowland Scotland ; the Scots language is a variant of Anglo Saxon with Scandanavian influence and tangential Gaelic and French influence.

Genetically the peoples of the British Isles are closely related and the dominant genetic strand (by far) is native as far back as the mesolithic ; the Scots have some Scandanavian genetics ,the English have significant Scandanavian and German genetics (especially in the east of both countries), the Irish have a strong genetic connection to Iberia on the male line only (this is all from Blood of the Isles).

The question about material culture is an interesting one. I would suggest that we can see clear connections between the material culture described as Celtic all over northern Europe and contemporary and near contemporary Ireland and Britain; but perhaps this is all in the eye of the beholder if the Irish had been designated Germanic perhaps we'd see the amazing connections to Germanic artefacts in pre Christian Irish art. When almost all Irish people see a Sile na Gig they see a Celtic artefact but these are found all over Europe. It is perhaps particularly non problematic to assert this continuity in the case of Wales as the Welsh were the unassimilated Britons (the Romans established forts in Wales but not much else) and the Britons are a clearly Celtic people (insofar as that word means anything more than saying that the Portugese, Mexicans, French and Romanians are Latin peoples).

Musically how are we to know? The music designated Celtic has strong commonalities but much of this is because they all listen to each others music.

However even if the Irish are not a Celtic people (and they may be as Celtic as anyone else) the notion that a Celtic consciousness is derived from some subaltern appropriation of an English Anglo Saxon consciousness is quite wrong.

The Four Masters assert that the Irish are descended from the Celts who are descended from Japhet who is Noah's son ; so the notion of Celticness at least in Ireland dates to at least 1616.