There is a bit of lighter colour on the fin edges, but I am not sure it's rot. If it is, it would be unusual for it to show on all the fins.
I've seen Bettas rip their own tails off, as it happens, from stress, in the case I dealt with. But it was growing back.
If you are not adding salt to the water, I would add some aquarium, sea or kosher salt, and so long as the water is kept very clean, he should be ok.
Keep an eye on him, but I think he is probably ok at this point.
The slight unevenness of the fin edges is not, of itself, a sign of rot, unless they were perfect when you first got him and have since become ragged looking. The fins move in a way that makes it hard to tell sometimes if what you see is just the effect of the fish moving, or if the fin is in fact damaged.
Just out of curiosity, what kind of substrate does the tank have ? I had one Betta that lost a lot of fin due to a coarse, chunky substrate that rubbed the edges of the gill and ventral fins. They grew back once he was taken off this substrate.. to a bare bottom tank. Any reasonably smooth textured substrate should be ok, just nothing with sharpish edges or corners.
