Is there some result that says a (sufficiently strong) theory cannot prove the consistency of any of its extensions?
Or something along these lines??
More generally, is there a result that says a (sufficiently strong) theory cannot prove the consistency of any stronger theory, for some notion of strength (proof-ordinals?), maybe with some conditions?
Yes, this follows directly from the 2nd incompleteness theorem.
By definition, a theory is inconsistent iff it contains both the statement $S$ and $\neg S$ for some statement $S$.
An extension of a theory contains all of the statements of that theory. Therefor if the extension is consistent, then the original theory is as well, since any such statements $S, \neg S$ in the original theory are also in the extension.