It seems that there are really more than two Elven languages, those are just the two in regular use in the areas we're concerned with. And that, even though Quenya is based on Finnish and Sindarin on Welsh, i.e. entirely different language groups in the real world, they really are supposed to be closely related and both derived from a common ancestor. In terms of vocabulary that's more or less plausible, with enough words that are similar in both languages, but in terms of grammar it doesn't make any sense. I suppose Tolkien realized that very well but didn't care because he found it far more exciting to develop two grammatically very different languages?
The way I understand it now from those Silmarillion passages, the original common language of the Elves at Cuivienen gradually evolved into different languages wherever Elves were separated - so not only a split between the Sindarin of the Sindar and the Quenya of the returning Noldor, but also the unnamed languages of the Teleri on Tol Eressea, as you mentioned, and of the Avari in the east. In the part about Finrod's encounter with the first Men, the text says that Men had learned most of their language from the Avari and that 'since all the languages of the Quendi were of one origin', this language of Men resembled Sindarin (as well as Quenya) sufficiently to make language acquisition easier. It doesn't say that those Men already spoke Sindarin, or Quenya.
As long as you accept the implausible point that they're all descendants of the same language - with Quenya also having deviated over time from the original united Elven language, even if it kept the same name - it sort of works. But yeah, it does depend on what assumptions you make about language evolution in a hypothetical world where people are immortal or nearly so. Clearly our languages do evolve even during our lifetimes, or within a few generations; for instance, Hindi and Urdu were considered the same language only a few generations ago, but nobody would still say that now. So if we lived hundreds or thousands of years ourselves, our languages would change to a certain extent from what we spoke when we were young. But surely indeed a lot less than the level of change that you see when there are dozens or hundreds of generations across that timespan, like in the real world. Especially for a people as conservative and nostalgic as the Elves.
You're right. And it literally says that the Avari lost contact with other Elves until 'many ages' later.
Yeah, apparently I was wrong on that, see above - it literally says the opposite, that all Elven languages are related and so knowing one of them helps with the others.
To add to this after checking the book - the language of the Men of both Gondor and Arnor is ultimately based on Elven, going back to that scene where the first ancestors of the Edain encounter Finrod. The language of the Eastern and Southern Middle-Earth Men would be based on never having encountered Elves at all. Also, we both forgot to point out that between Gondor and Dale/the Shire, you do have another kingdom of Men which does actually speak a different language, Rohan. Or at least, while they also speak the common 'Westron' of Gondor and the Shire, this is a second language for them and in the Rohan chapters you regularly see references to words in the native Rohirrim language (which is based on Old English).