Zero monomorphism In a category with zero morphisms, can someone think of an example where $A\rightarrow B$ is a zero monomorphism but $A$ is not a zero object? 
(It is easy to see that $A$ should be a terminal object.)
 A: Consider the category of sets and let $A = \lbrace a \rbrace$. Certainly, the category of sets has zero morphisms. However, $A \to \lbrace a,b\rbrace$ is a zero morphism sending $a$ to $a$. But $A$ is not a zero object.
A: If your category contains zero objects, then all terminal objects are initial, and hence zero objects. (And dually, all initial objects are terminal.)
Proof: Let $T$ be a terminal object, $Z$ be a zero object and $X$ be any object. There are unique arrows $T \overset{i}{\underset{j}{\rightleftarrows}} Z$ and these must form an isomorphism since the identities are the unique arrows $T \to T$ and $Z \to Z$. If $X$ is any object then there is a unique arrow $f:Z \to X$, but then there is an arrow $fi:T \to X$, and if $g:T \to X$ is another such arrow then $f=gj$ and so $fi=gji=g$. Hence $T$ is initial. $\square$
So, to half-answer your question, if $A \to B$ is a zero monomorphism in a category $\mathcal{C}$ then, since $A$ is necessarily terminal, $A$ is a zero object if and only if $\mathcal{C}$ has a zero object. So to find the example you seek, all you have to do is find a zero monomorphism in a category without a zero object.
A: By "category with zero morphisms" I will assume you mean a category enriched over pointed sets, so that there is a distinguished zero morphism between any two objects. Suppose $0_{A,B} : A \to B$ is a monomorphism and also a zero morphism. Then,
$$0_{A,B} \circ \textrm{id}_A = 0_{A,B} \circ 0_{A,A}$$
and so cancelling $0_{A,B}$, we get $\textrm{id}_A = 0_{A,A}$. It follows that $A$ is a zero object.
