Alon with bullsnakes, they're extremely useful for getting rid of said rodents—which CAN carry awful pathogens, like hanta virus!
Fortunately, snakes will generally leave you alone unless they're provoked or cornered. Unfortunately, if you have curios animals (cats, dogs) it can be a much more significant problem!
So that's good for the birds, and bad for the humans that want to get rid of all the pesky annoying mosquitoes, not just get rid of mosquito born disease.
In that case, not even need to exterminate any species.
That's typically done by introducing some Wolbachia in their gut.
The three on my "eliminate at all costs" have always been mosquitos, fleas, and ticks.
But yeah, mosquitoes and cockroaches should be made extinct, even if they are tough critters.
Talking of tough though..
Say hello to the only creature that's evolved to cheat death itself: the Immortal Jellyfish. https://en.wikipedia.org/wiki/immortal_jellyfish