It's a very similar construction in Irish.
Tá cat agam = A cat is at me (literally).
No, Fi is the base word and it sometimes is said as "i" because it is easier to say.
Yes, in more formal registers though typically courses teach "fi" as the starting point in order to simplify things.
It's "Mae gen i gath". Literally "a cat is with me", in that sentence "mae" means "is".
- Mae gen i gath
Mae cath gen i
Mae gyda fi gath
- Mae cath gyda fi
Yep. Gen is northern (and it changes form depending on person) and gyda is southern (gyda doesn't change). The order just changes the emphasis.
I'd recommend you check with your tutor about that, the only third person singular form of to be is "Mae" with dialect pronunciations of "Ma"