Screwworms could probably be eradicated in theory but it would require spreading the sterile fly program to the entirety of the Americas which isn't going to happen. There would always be a pocket somewhere in the Amazon of fertile flies so it isn't really viable. The point of stopping them at the Darien Gap was that there was a geographically small area where their spread could be halted from entering Central and North America and re-establishing themselves.

It depends on the pest. Some of them are easier to eliminate than others. With screwworm flies the only offense we have is to raise them by the billions, sterilize them with radiation, chill them down, and then drop them out of airplanes. Fertile females end up mating with sterile males and then cannot lay any eggs before they die. Each generation then becomes radically smaller than the previous. Since their lifecycle is only a few weeks long this eliminates them in a few months. They were able to successfully eradicate the screwworm fly from North and Central America, but a combination of expense and diplomatic entanglements prevented them from continuing south past Panama. There have been outbreaks before, most notably in Egypt (or maybe just northern Africa, I forget) a few decades ago.

We have different responses to other pests. For example, Florida maintains a mosquito control program that sprays vast swathes of the state with insecticide from both the ground and the air every 7 days. I imagine that other southern states do as well.

Rinderpest (a cattle disease) and Smallpox are the only two diseases ever successfully eradicated. The smallpox vaccine was the first vaccine ever invented and it took until 1980, about 180 years later, to eradicate the disease entirely. It pretty much is an unceasing war, though Guinea worm and polio are also relatively close to being eradicated. But if you stop fighting them, they'll just spread again.