In this context, it means "steal." It's fairly common to use it that way, but it is very informal, almost slang. It suggests either a small theft, or to take something in a sly, sneaky, or surreptitious way. "When the teacher was out of the room I was able to cop a peek at the exam..."

It's fairly common American slang. I don't think it necessarily mean to steal, but it does give the feeling that you're trying to get something that may be hard to get and you need to act quickly.

In the UK, no. I've never actually heard that word used in that precise context before (in the sense of being about to possess something.)

It's slang, used in place of 'buy'. It's popular in online fashion forums, among teenagers and people in their early twenties.