It's not used in the United States-- --but the expression "give someone a RING" is. For example, "Why don't you give me a ring tomorrow?" Also, "ring someone up." "I've been trying to ring him up all day but he doesn't answer."

It is a very British term, some Americans might understand and use it, but I do not think it is a well known phrase in the USA.

In fact, 'give someone a bell' is not often used in the UK either. It strikes me as quite old-fashioned. It reminds me of films from the early and mid 20th century, when phones had actual bells attached to them. I would never say this, and I can't recall ever having heard anyone say it in recent decades. We tend to say 'give someone a ring', as in the US, or 'give someone a call'. And to answer your second question, yes, 'ring someone up' is a common expression, at least in British English. If you 'ring something up' you enter the product on a cash register, and if you 'ring someone up' this means that you phone them.