This is a good question because this stuff is confusing. There are also multiple correct answers...
1. The key is E major (4 sharps in the key signature).
2. Correct, in the key of A major, ii is B minor.
3. You can think of it as a modal thing... A Lydian. All the chords can be found in E major without raising or lowering any pitches. In other words: E major = A Lydian.
It is a nice chord progression, especially on a 7 string guitar.