Its a bit hard to tell from the photos if he does have fin rot. But if he does its not bad at all. Fin rot is usually characterized by fins that look shredded and clear and your fish is losing more and more of his fins so it gets closer to his body. But it looks like it could be his fins growing out as well. Sometimes with younger fish when their fins are still growing they have black on the very edge it could also be new coloration. Could you get on of him flaring or at least with his fins open so its a bit easier to see what his fins look like?
A couple of suggestions.
I would switch his food. Tetra betta food tends to contain a lot of fillers and gets your tank dirty pretty fast. The two most recommended brands on this site are Omega 1 Betta Buffet Pellets and New Life Spectrum pellets for bettas. You should be able to find either one at petco, and their good food for your fish
.Plus at least with the omega one since I haven't tried that other one, a small container will last you a long time especialy with just one betta. I would also feed him a bit less. Betta's have very small stomachs but unfortunetly are little piggies so they will eat and eat. Once a day should be fine even every other day will work well for bettas.
My other suggestion is that I would do less water changes. If your tank is cycled then once a week will be sufficient for the tank. To many or to large of a water change can mess with the cycle of your tank.
Also quick question, how long has your tank been running before you put him in and how have you cycled the tank, just to make sure that it is for sure cycled.
A couple of suggestions.
I would switch his food. Tetra betta food tends to contain a lot of fillers and gets your tank dirty pretty fast. The two most recommended brands on this site are Omega 1 Betta Buffet Pellets and New Life Spectrum pellets for bettas. You should be able to find either one at petco, and their good food for your fish
My other suggestion is that I would do less water changes. If your tank is cycled then once a week will be sufficient for the tank. To many or to large of a water change can mess with the cycle of your tank.
Also quick question, how long has your tank been running before you put him in and how have you cycled the tank, just to make sure that it is for sure cycled.