You're misunderstanding something. P=NP is about whether two sets are equal. If it's "undecidable" or "independent", then ZFC is consistent with either P=NP or P \neq NP.
Independent doesn't sound like the same thing as undecidable.
Given a program P, it is undecidable whether that program will halt. There is an answer: P either halts or it doesn't. There is just no way to find out (in general).
Yes, 'undecidable' is used for a similar class of mathematical problems. For example, the continuum hypothesis - another question about an equality of sets - is undecidable in ZFC. This notion is strongly related to undecidability on Turing machines (possibly equivalent, depending on how you look at it).
They are intimately related. If you look at how Gödel's incompleteness theorem is stated, the existence of "effective procedures" (I.e. Computable functions) figures in intimately.
Scott Aaronson is probably one of the best at explaining theoretical computer science, if you're interested in these topics you should follow his blog.
Given a program P, it is undecidable whether that program will halt. There is an answer: P either halts or it doesn't. There is just no way to find out (in general).