I will bet you money it is the neons. Neon tetra are schooling fish and need to be in groups of 6 or more (preferably more than 8 individuals). Without sufficient numbers they will often lash out at the other fish in the tank, and betta are usually a prime target. Frankly, I would remove the betta to a 3 gal tank and keep the cories and neons in the 10 gal tanks. But you're going to want to increase the number of neons or else they could turn on the cories.
For fin regrowth, the best thing is clean water. Aquarium salt just treats finrot and won't actually help regrow the fins. Warm, clean water and good food will regrow them.