Let $I$ be initial in some category $\mathsf C$. Then, if $X$ is a second object of $\mathsf C$, it exists and it is unique the morphism $f: I \to X$.
So let $g$ be an endomorphism of $X$ different from $\mathrm{id}_X$. This implies that $g \circ f$ is a second morphism $I \to X$, then it must be the same as $f$, by initiality of $I$.
What does this say about the endomorphisms of the objects of $\mathsf C$?
In a precise sense, it tells you absolutely nothing. Let $C$ be an arbitrary category. We can form a new category $C^+$ by adding a new object $o$ and for each object $x\in C$ a unique morphism $f_x: o\rightarrow x$ (note that there is now exactly one way to make sense of composition). Now in $C^+$, the object $o$ is the unique initial object ($C$ itself may have had an initial object already, but it's no longer initial in $C^+$). But intuitively, $C^+$ has all the "variety" of $C$ present in it (formally: $C$ is a full subcategory of $C^+$). So having an initial object doesn't really cause any restrictions at all.
It may be worth thinking of Set in particular. The initial object is $\emptyset$, and the unique morphism $e_X$ from $\emptyset$ to $X$ is the empty map. If you think about this example, I think you'll understand why having an initial object doesn't really tell you very much about arbitrary endomorphisms in the category (basically, composing with the empty map "trivializes" things).