I'm also American, and I was surprised to see your comment. I wonder if this depends on where you live, like "soda" vs. "pop." I and most people I know would say either "do you have a temperature" or "do you have a fever." They both are equally common where I'm from. I can't imagine asking "do you have a high temperature?" though. That is strange.
In Ukrainian, the proper proper way of saying it is also "висока температура" (high temperature) or "гарячка" (fever). However, the peculiar "У вас є температура" (without висока) has entered the scene probably as a colloquial version, and is now considered more or less standard. This is my current understanding :)
This sentence reminded me my childhood. Basically, everybody has a temperature, except for a dead body out in the freezing weather. This is one of those sentences we made fun at as children. One child would ask another one, "У вас есть є температура?" The other child would say "No" and drop down, pretending to be dead and having no temperature at all. The right question about somebody's health would be if their temperature is normal, above normal or under normal. The translation "Do you have a high temperature?" makes sense to me but the Ukrainian sentence presented here sounds funny. Exactly because every body has some temperature, except for in the case described above.
Questions typically do use "є". Both are kind of possible, though, but I feel the slight difference in meaning.
- "У вас є температура?" is a very neutral way to ask the question, you just want to know if someone does or does not have fever.
- "У вас температура ?" is more of "Is it the fever that you have?" (or some other thing)