The thing that makes Aedes aegypti such a good vector is its tendency to feed on multiple humans. They get spooked and fly off from a feeding site, then land again to try again, and the subsequent bites might not be on the same human. So one infected mosquito can vector a disease to an entire room of people if there is enough activity to make them attempt that many bites.

Many other mosquitoes will just stick it out and hope you don't notice them before they fill up. They either succeed in one bite or get squashed. Those mosquitoes have less opportunity to acquire a disease from an infected human, and less opportunity to transmit one to an uninfected human. And if there is no mechanism for that vector to pass to a mosquito's offspring, the one-bite behavior means there is no transmission between humans at all.

I expect that would depend on why only certain species can transmit human malaria. Yes if the reason is purely regional, but if there's something else (length of lifecycle, presence or absence of certain enzymes, ability to take blood from human) then there would have to be incredibly fast adaptation for malaria to survive.

The parasites don't survive equally well even within the species they inhabit.

https://www.cdc.gov/malaria/about/biology/mosquitoes/

Probably not. Humans have eliminated many species and the niches they occupied haven't been always been filled by other species. Megafauna on many continents is a good example.