The Lazarus Experiment
He plays the organ, after which his hearing is temporarily affected.
I don't think it really was. I always see this scene as him joking with Martha, because he doesn't want to answer the question. But maybe that's just my version.

Only other thing I can remember is, that they evidently have some sort of normal childhood till they're eight and get shown this time vortex thingy. Maybe they only change into timelords afterwards at school/academy?

And yeah, the mind melt.

And about the regeneration question: Maybe the 'rewriting cells' only works if you rewrite all of them at the same time. You know, change the whole being so it fits again. If you only rewrite parts you have a different code (DNA) in different bodyparts in the end, and that wouldn't work. So maybe that's too, why he can't just heal himself. Because either you do the whole thing, or you don't do it at all.
Also, I think healing yourself and rewriting your biology are two different things, no? So maybe he can only do one, but not the other.