What is your definition of measurable? I assume you mean that

\(\displaystyle f^{-1}(B) \in \mathcal{F}\)

for each Borel set.

So here are some hints;

First show 2) by considering sets of the form [a,\infty) then show 1 by using the 2 complement and sets

\(\displaystyle

\{f \geq n\}\)

For 3 consider sets of the form \(\displaystyle [a-1/n,a+1/n]\).

Q2 is wrong, you need the function to be surjective. You should attempt it first (and post here).