Well if you really want his fins to heal nicely, do a water change every day, and add aquarium salt. If you're using bettafix just for the fins, since I don't have a picture I don't know how "necessary" that is (but really, IMO, medication is never necessary for fins. But thats only me). I also feed my betta fish their pellets soaked in garlic juice and they always heal just fine with a torn fin.
Okay, I just read that again. He's a nemo - it probably was either physically damaged, or birth defect as a fry. I would imagine it heals at the same rate as fins, but who knows. Definitely watch for that. Given the size, I would give it a week.
IMO, about temperature, he will look fine, but 75 is slightly pushing it temperature wise. And I would imagine it gets cold at night, just being in a house, too. I live in Texas, so I know warm weather when I see it, haha. But my house drops to 72-73 at night, and the temperature fluctuation, and the plain temperature, is not optimal for them. I always say 77-80F, with 75 pushing it. And also, lower temperatures seem to affect their immune system, if ever so slightly. If you can get a heater, definitely get one.
If you really want to take care of him, how about you invest in a larger tank? I prefer 2.5 gal absolutely minimum, 5 gallon highly preferably. You may hear other stuff, but this is just my own opinion, and I wouldn't house a betta under 5 gallons (I have a 5 gal divided, but I'm never dividing that again).