No, it isn't. A 10g is too small. Goldfish are sociable creatures therefore you cannot keep only one. A 40g is my recommended starting point if you want to try fancies. It can fit at least 2-3 goldfish. A lot of fancy goldfish are bulky growing at 8-10 inches. My oranda is 7" and that's still not enough for him. Aim for 10-15g per fancy goldfish but that doesn't mean anything smaller than a 30g is feasible because it is not.
Yeah you can use betta fix with the frog, It may effect his appetite a bit though. as far as keeping him calm do not do any that would cause him to flare. He wil anyway, just no extra.
It may or may not, It will start growing back,some betta get over excited though and split them again. Try and keep his excitement level down until he heals.
Bettas are usually very healthy and hearty fish. They are prone to fin rot, but clean water and salt baths when caught early usually remedy that.
Alot of it depends on where you get it from too. So many pet stores have bettas in such poor conditions that there is no telling what issues they might have. Always look for a healthy active one!