What's the finish line with the Doom of Damocles? Is it when the apprentice becomes a full wizard?
...
What has Harry agreed to and what level of responsibility comes with that? For example, if Harry agrees to teach Kim enough to control her powers so she doesn't accidentally cause harm, what is he responsible for? Would it be a different level of responsibility than if he agreed to teach her some stuff about magic?
Depends... Harry was under the Doom until the Council lifted it. He had been considered a full wizard (albeit a disreputable, kitty-literbox'es-his-dress-robes sort of full wizard).
Then he went back under the Doom, effectively, when Molly did (if she had gone Warlock, he'd have been subject to being executed alongside her). She's
got to be out from under, as the new Winter Lady, so he is too.
I suspect that if Molly had gone warlock, Harry could have escaped execution by turning her in (with the recommendation to kill, instead of clemency); or maybe if he killed her himself.
I presume Eb was under the Doom, too, while Harry was (and, egads, can you just
imagine the consternation amonst the Wardens if they had been told their target was McCoy the Blackstaff??!?); that sharing the Doom is part of rehabilitating the proto-warlock.
I recall (but cannot cite, sorry) that Eb could have escaped the Doom by just killing Harry outright: "He was just too far gone, I tried but I couldn't fix him." That's why/how I think Harry could have gotten out if Molly had gone bad.
BUT...
... What has Harry agreed to and what level of responsibility comes with that? For example, if Harry agrees to teach Kim enough to control her powers so she doesn't accidentally cause harm, what is he responsible for? Would it be a different level of responsibility than if he agreed to teach her some stuff about magic?
I don't think the sword of Damocles applies to the Kim situation...
Agreed -- Kim wasn't under the Doom (so neither was Harry) but simply because Kim wasn't a White Council wizard (she was too weak a talent for the WC). The "Doom" is a privilege, so to speak. Minor sorcerors &c just get killed for warlock'ery. Harry would have been obligated to kill her (or report her up the chain) if he'd been greycloaking then, but before the cloak he was an "ordinary" WC member, and (afaik) under no special obligations (other than NOT to aid/assist with warlock'ery).