English spelling is infamously irregular. Is it possible for it to branch into several categories (e.g., Germanic spelling, French spelling, Greek spelling, etc.)?

Yes indeed. Bear in mind in particular that Greek and Latin fall under the rules of Traditional English pronunciation of Latin. (Greek is almost always borrowed into English via Latin; but there are late exceptions like kudos, not †cydus.) Those rules are not the rules of French words in English. For example, final –e in a Greek word like psyche is always pronounced; it is never pronounced in French or Germanic words.

(That link should be mandatory reading for everyone ever btw. There’s a lot of stuff I’ve learned from it.)

Germanic and French spellings get squashed together via the crucible of Middle English spelling—and Middle English itself; but you can still discern differences. For example <k> is a Germanic thing; word-final stress is a French thing.