Learn English – “To mentor someone during a project” vs. “to mentor someone on a project”

"John got a mentor to guide him throughout his life" implies that the act of acquiring a mentor was a single event that took place in the past. It says nothing about the length or status of John's life in the present.

"John got mentored on a daily basis", taken in isolation, implies that John is no longer being mentored on a daily basis; the mentoring stopped at some point in the past. (In a story being told in past tense, however, it would not necessarily carry any such implication.)

Taken in concert without further context, the two statements form a tiny story told in the past tense and can be read either way. If you want to make it clear that the mentoring is ongoing, you would probably want to switch tenses and say "Now he gets mentored every day." However, the use of the past tense only implies, at most, that the mentoring has stopped, not that John's life has stopped. (Perhaps he couldn't afford the mentor, or perhaps he learned everything he wanted.)

Yes, they are both grammatical, but With what are you doing that? is much more formal, and probably quite rare.