Yes, some shelter are better than others. No argument. But, shelters are not the sole reason for high euthanasia rates. As Mark pointed out, there are regional trends, and in some areas, there are simply more unwanted animals than homes can be found for, despite the efforts of shelter personnel, volunteers, and rescues. I get several emails a week from shelters in AL and some areas of FL asking me if I have room for some of their dogs. Most of the time, I have to tell them that I don't have room. And it is the same with many other rescues. We are full all. the. time.
No, it's not. It's a pretty standard, first step question to ask. Most shelters are rescue-friendly. A few are not. The ones that are not are usually not because they'd been burned by rescues (usually out-of-state ones) in the past, or they don't need rescue's help because they have good success placing the animals on their own.
Who, exactly, are these rescues that are importing dogs from other countries "by the thousands"?