Sure. But I think it's very fair to say that Irish fairies are far older than any recorded Germanic tradition of elves.
I can't say I agree. I will say that the only flaw I can find in Tolkien, whom otherwise I regard as having a sacredness just below Scripture, is that it feels strange that his world is so empty of religion. However, I don't think creating a Christian mythology is in any way strange. He wasn't trying to found a new sect of neo-paganism.
Irish mythology is old. It was written down by monks, yes, but it remarkably retained its pagan nature. Yes,
Sidhe is vague;
Tuatha De Danann is not.
Worth noting that Tolkien absolutely detested those.
Not off the top of my head--I'm thinking it was a Brythonic source--but if I find it I'll let you know. I think I ran across it when I was doing research for a paper on
Sir Gawain and the Green Knight.
If you don't have experience with literary criticism or Anglo-Saxon studies, you have no idea what a bold statement you just made.

There are a number of theories about
Beowulf: it was an oral story written down by a monk; it was a bit of historical fiction written by a monk; whether original or traditional, it was written by a pagan who had a knowledge of Christianity (if you're familiar with the tortuous history of Christianization in England, that's not so strange); among others. (For what it's worth, I believe you're correct, that it's a traditional story--or more than one--that was recorded by a Christian monk. But I'm not an expert, and the topic is still hotly controversial.)