Maybe I misunderstood the lessons on pronouns, but since "I" am the one finding the keys and they're "my" keys, shouldn't we use swoje instead of moje?
The meaning stays the same, however "swoje" is the better option, stylistically. Both should be accepted.
Is klucze accusative and kluczy genitive?
Yes, it is. And „klucze” can be also nominative.
Could moje be omitted since the subject is also 1st person?
Yes, it seems most likely that those are "my" keys anyway. A version without "moje" works.