Proof of (1) is basically correct except for some typos:

This is where you began, what you proved is that . Also, is not equivalent to

, but the implication from left to right holds, and that is what you need.

(2) is injective: apply (1) to is injective.

is surjective: if it is not, then also isn't surjective, but is surjective.

Further, we have , because composition of functions is associative.