The whole point of the Laws of Magic with regard to killing people, messing with their mind, compelling them, is to prevent the practitioner from being subject to black magic creep which damages their mind.
To a large degree, but I think it's important to distinguish between "the Laws as enforced by the White Council" and "what is actually corrupting". The first is highly subject to politics, the second presumably isn't.
(And I'm not sure that preventing the corruption is the only motive of the Laws of Magic either. The White Council is also supposed to be about restricting the power of wizards over mortals. Things like time travel and necromancy may not be corrupting in the same sense, just really dangerous to mess with.)
- Is Harry protected now that he is a Knight? Is he immune to the concerns about black magic eating at his soul?
I strongly doubt it. Mab doesn't care about souls, and Winter Knights apparently don't last that long usually.
- To what extent does the breaking of the laws corrupt the mind? Are some more dangerous than others?
We really don't know. It seems like there probably is a difference, though - Cowl and Kumori seem pretty controlled and 'sane' compared to other warlocks we've seen. (Their
goals are crazy, but they seem quite capable of acting toward those goals in a highly competent way.) Necromancy may not be corrupting in the same way that killing or mind magic is.
Can the corruption be "healed" with time or is a permanent stain on the soul?
I think so, at least if the person acts against the impulses that are associated with breaking that Law. Harry really doesn't seem to be meaningfully tainted by breaking the First Law any longer, IMO.