I couldn't see any rot in the pictures, but the pictures weren't that clear so was hard to tell exactly.. he does have a case of biting though- or did when he was in the cup.
Forget about the salt at this time, keep up with the weekly water changes and he should be fine.
Clear-white coloration at the edges is new growth, but keep an eye on them. Look for blackened edges that seem flaky.. that is rot. But if the edges are smooth and the natural color then he is biting.
Bettas will change colors once they settle into their new home. He changed to darker/blander when you took him out because he was stressed.
Good luck and if you have any more questions, please feel free to ask.