Difference between Stabilizer and Centralizer?

As a preliminary, you cannot talk about stabilizer without specifying an action.

(1) Yes.

(2)The centralizer in $G$ of $a\in G$ is the stabilizer of $a$ for $G$ acting on $G$ by conjugation (i.e. $g.h:=ghg^{-1}$). So the centralizer may be a stabilizer, it actually depends on the action.

(3) Yes, if you choose the good action.

(4) If you are still considering the action of $G$ on $G$ by conjugation then $Z(G)$ is the kernel of the action (it is also the globally fixed points set). This implies ( because each stabilizer contains the kernel of the action) $Z(G)\subseteq C_G(a)$.

You should start off by answering question #1 yourself; this is a great collection of questions, and you're definitely capable of it.

Every group $G$ acts on itself by conjugation; $x \mapsto x^g = g^{-1}xg\ $ (prove this). Under this action, you should figure out what the more common names for "orbit" and "stabilizer" of a group element are. This will show you how centralizers and stabilizers are related. Note that the relationship is that of "proper containment", as there are other ways a group can act on itself besides conjugation (although conjugation is a very special action). This should also shed more light on question #3, if you know a certain famous theorem about group actions.

For bonus fun, let $\cal{S}$ be the set of subgroups of $G$. See what happens when $G$ acts on $\cal{S}$ by conjugation; do stabilizers or orbits of subgroups have any more common names?