Undecidable vs Unsolvable?
Yes, undecidable also applies to problems that cannot be represented by TM at all. The ones where a YES can always be found are a subclass of these called (recursively) enumerable. Unsolvable is a less common term which I know mostly to be used synonymous to undecidable.

So  decidable - undecidable but enumerable - non-enumerable  is a partition of the class of all problems into three classes, where the middle one is intuitively somthing like the easiest undecidable problems.