Yes. the set of rational numbers is a countable set which can be generated from repeatedly taking countable union, countable intersection and countable complement, etc. Therefore, it is a Borel Set.

Yes.

all finite set is countable.but,countable can be finite or infinite

No, it is uncountable. The set of real numbers is uncountable and the set of rational numbers is countable, since the set of real numbers is simply the union of both, it follows that the set of irrational numbers must also be uncountable. (The union of two countable sets is countable.)

It is NOT a 'countable set'. It is an infinite set. 1, 3, 5, 7, 9, 11, ... you can count to infinity and keep going.