It’s a little imprecise. ‘Handling’ implies something more complicated than simply paying. Natural: I paid/picked up/got the check.

Yes, the same in South Africa and the UK.

It makes so much sense that you should not worry about whether or not it is commonly used. "Handle" is especially useful if the check is somewhat complicated. If it requires any sort of thinking beyond just pulling out a credit card, than "handle" would be better than the other choices.

I agree with Dan that "handled the check" isn't heard often and "took care of the check" is more common. I would add that "picked up the check" is probably used even more frequently.

Yes. It's very natural. It's not especially common. I think the most common phrase in the US would be "Sara took care of the check." There are a lot of ways to say it. Sometimes people do not like to talk directly about money, so saying "Sara took care of the check" or "Sara handled the check" would be a little softer than saying "Sara paid the check."