(Disclaimer: I'm nowhere near truly knowledgeable about any of this)
Quantum computing was explicitly proposed as a way to prove the truth of many worlds. As far as we can tell, quantum computing appears to work.
I don't think most people believe QC truly proves many-worlds, but I think it is considered some evidence in its favor, and historically it was I think one of the reasons for the development of QC originally.
I don't think there is anything in quantum computing that is specific yo the many worlds interpretation.
I've heard this claim, but I don't think it holds up. The idea is that supposedly the calculation is done in parallel across the different universes which allows you to see an exponential speedup in some calculations. But the whole point of many-worlds is that after a branch event the various branched portions of the wavefunction no longer interact. Even if it were true that quantum computing was really a bunch of parallel classical computations (which it is not), there would be no way for any of the other universes to communicate the result back to the universe you observe it in.