I left GB nearly 30 years ago and what I still find strange is when you leave a till at the supermarket (in GB) they don't say Bye anymore but See you! as though they're my friend. I probably won't see them either...
Goodbye is just a way of saying "End of conversation". (It seldom really means "God be with you", although that seems to be where it comes from.) "Thank you" will often have the same meaning. I don't think it's rude.
i forget sometimes, but my mum goes bye, byee, bye, and then hangs up the phone, it's funny sometimes, and i reckon a lot of people do it without realising.
I usually say 'see you later' in the shop. Most of the time you find that you have the same customers every week at the same time so you kind of become a little bit familiar with them.
Now that I think about it... i'm going to miss all my customers like that! :-(