1) Assume that $0 \rightarrow A_i \rightarrow B_i \rightarrow C_i \rightarrow 0$ and $0 \rightarrow C_1 \rightarrow C_2 \rightarrow D \rightarrow 0$ are exact, $i=1,2$. Show, using a diagram chase, that $0 \rightarrow B_1 \rightarrow B_2 \rightarrow D$ is also exact.
Let $i_j : A_j \rightarrow B_j$, $p_j : B_j \rightarrow C_j$, $\alpha : B_1 \rightarrow B_2$, $\beta : B_2 \rightarrow D$. There's a commutative diagram with this indicating that $\alpha i_1 = i_2$ and $jp_1=p_2 \alpha$. I don't know how to this. I figured there's a $b \in B_2$ s.t $qp_2(b)=d \in D$, but I've no idea how to continue, or if that's even useful?
2) Let $0 \rightarrow A' \rightarrow A \rightarrow A'' \rightarrow 0$ be an s.e.s. of abelian groups. Show that the following are equivalent:
i. There exists a homomorphism $j: A \rightarrow A'$ s.t $qi=1_{A'}$.
ii. There exists a homomorphism $q: A'' \rightarrow A$ s.t. $pj=1_{A''}$.
iii. There exists an isomorphism $\phi : A \rightarrow A' \oplus A''$ s.t. $\phi i$ is inclusion into the first coordinate and $p\phi^{-1}$ is the projection onto the second coordinate.
I haven't the slightest idea??
3) Advise on what to do when you can't even answer the questions on the first homework? Except pulling your hair out by the roots, I mean...
1) You missed something in your question: if you choose $C_i=D=0, {\rm Hom}(B_1,B_2)=0, B_1\ne 0$, you will not get an exact sequence $0 \rightarrow B_1 \rightarrow B_2 \rightarrow D$.
2) You can find a proof in any textbook on homological algebra (e.g., S.Mac Lane, Homology).