II.1.20

This problem has Hartshorne Height 2. II.1.2 can be used to solve this.

HAPPY

 * Part a) just requires the zero and gluing axiom; and important point is $$ 0 \in \Gamma_{V\cap Z}(V,F|_V)$$ is equivalent to a section whose support is empty. The gluing axiom from the fact that $$F$$ is already a sheaf, the only thing to be careful is to make sure the elements you get by gluing have support contained in $$Z$$.
 * Part b) can be done by using II.1.2 and showing exactness on stalks.
 * Finally for the last statement, note that the stronger condition that for every $$U$$, a sheaf map $$F(U) \to G(U)$$ is surjective, implies that the map $$F \to G$$ is surjective.