Let $C$ be an abelian category (maybe, plus some nice properties.)
Let $X$ be an object of $C$ and let $i:Y\to X$ be a (representative of) subobject of$X$.
Question 1: Is there a projection $p$ from $X$ to $Y$ so that $p\circ i=\mathrm{id}_{Y}$?
I came across this question when I tried to prove that $X$ is a direct sum of $Y$ and $X/Y$. (I am not sure if this is true.)
Question 2 If an object $X$ has a subject $Y$, is $X=Y\oplus X/Y$? How do I make sense of $X/Y$ in an abelian category? Is it the cokernel of the monomorphism of the subobject $i: Y \to X$?
The answer is no to both questions. When such a map exists, we say that $Y$ is a direct summand of $X$. If all subobjects of objects in an abelian category $C$ are direct summands, then $C$ is said to be semisimple, and this is a very restrictive condition; as its name suggests, if $C = \text{Mod}(R)$ is the abelian category of modules over a ring, then this condition holds iff $R$ is a semisimple ring in the usual sense.
In particular, $R = \mathbb{Z}$ is not semisimple. The subobject $2\mathbb{Z} \subset \mathbb{Z}$ is not a direct summand. Equivalently, the short exact sequence
$$0 \to \mathbb{Z} \xrightarrow{2} \mathbb{Z} \to \mathbb{Z}_2 \to 0$$
does not split, and $\mathbb{Z}$ is not isomorphic to $\mathbb{Z} \oplus \mathbb{Z}_2$. This reflects the existence of interesting Ext groups in this category.
You are correct that $X/Y$ is the cokernel of the inclusion.