He is just much better at 'thinking outside the box' - that is, to see solutions outside of their parameters.

Not really. He is indeed aware of a different box to earth-bound people, but the show has repeatedly represented the Doctor as very prejudiced against the military. It made sense as part of his reaction following the Time War, but it was still basically prejudice.

if they somehow rewind time to the moment when the Monks too over, would the Doctor still die?

I'm guessing if they do do a rewind they will fudge the lab error. Whatever happens, the Doctor can just nip back and tell that scientist to have fewer drinks and make sure Erica has a spare pair of glasses. Unless he can't find where Nardole parked...