On explaining monomorphisms and epimorphisms

This is fine. Your definitions (which are the contrapositive of the conventional definitions) are logically equivalent to the conventional definitions, and if you think your definitions better supports an intuitive understanding then you are free to use them.

I would add a footnote explaining that this is not the conventional way of writing down the definition, just so your readers are prepared should they want to read up on it on their own afterwards.

This is indeed equivalent, as you just take the contrapositive of the usual definition.

I do not agree that this would be better though. We usually use the fact that an arrow is monic or epic to prove that a diagram commutes. The usual definition fits that better. However, this is personal preference and your intuition might be different. Just something you may want to take into consideration.