I don't usually stand when someone enters my cubicle, but I've been wondering lately if I'm supposed to. I'm an intern so almost everyone who comes to see me is a supervisor in some way. Is it rude not to stand? Also, is it rude to offer someone a chair without standing first?
If you're being introduced to someone for the first time -- a new colleague, a customer, etc., you generally stand up and shake their hand. Otherwise, it's generally OK to remain seated, and you don't need to shake hands. The possible exception might be for members of senior management, if these are people you don't interact with frequently.
Gender roles have shifted over time, but it used to be the case that men stood up to shake hands, while women would remain seated (and gentleman would stand up when a lady entered or left a room). I doubt there's any such distinction anymore in the American workplace.
I'm a programmer, so I have worked with introverts who don't even turn around in their chairs when you enter their cubes, so there's a wide spectrum of behavior, and every office has its culture.
As someone who's worked in a rather small cubicle for the last 5 years, and a couple of years in a previous role, I have to say no, it's not rude. Nor do I expect my co-workers to stand when I enter their area.
The only exception may be if the co-worker needs to sit to type something on your computer, such as to install software or make some sort of configuration change or the like.
Is it rude not to stand? Also, is it rude to offer someone a chair without standing first?
In the US, in general, it's fine to remain seated when another employee enters your cubicle. Similarly you can offer a chair without standing.
On the other hand, if the person entering is a customer (say, you are an automobile salesman), then you should stand.
Most work in the US is rather informal these days.
If you aren't sure, just notice what others around you do. If you happen to work in one of the few companies that remains very formal and you see others standing whenever anyone enters their cubicle, then simply follow their lead.