Can I charge people gold for big requests like making full custom mugs? I expect not since checking whether people update their gold or not would be too much hassle, but I wanted to ask.

They've been doing it for as long as I recall. :P
Shouldn't be a problem, but wait for a mod just in case.

If they want to pay you gold for your work, it's really their problem for not being as good at mug making as you, right? Since it's basically a normal transaction of gold between two members, I see no problem with it.