"shall" is the same as "should". If you ask "shall I pay now", you are asking the listener if they think it is a good idea to pay now. The word "may" is used to ask permission or ask is something is possible. Examples: "May I borrow ten dollars?" "May I have another piece of pie?" Most of the time in casual conversation, people use "can" instead of "may", as in "Can I borrow fifty dollars?"

I agree with Erik. In most cases, there is no difference between 'Shall I pay now? and 'Should I pay now?'. They are both offers. There may be slightly more obligation/advisability implied in the 'should' question, but they are often interchangeable. As Hali says, you expect the answer to be 'Yes'. These are different from 'Do I have to pay now?'., which is actually an enquiry about whether is is necessary/obligatory to pay now. 'May I pay now?' is asking permission. This sounds a little unnatural, as we tend to use 'Can I ..?' rather than 'May I..? ' in everyday speech nowadays.

"Shall I pay now" is technically correct, but has slightly different connotations than "Do I have to pay now?" . "Do I have to pay now?" is a completely open-ended inquiry; it does not expect any specific answer. Whereas "Shall I pay now?" slightly expects the answer to be "Yes", because "Shall" implies definitiveness. When you say you "shall" do something, it is similar to saying you "will" do something. "Must I have to pay?" is not correct, because "must" and "have" are redundant. "May I pay now?" is asking permission to, not whether you are going to or not, or whether you can or not.

Shall IS NOT the same as Should. Shall is more definite than should. Meaning Should is questioning permission instead stating that you are. May would more so used in referring to paying something although that question is not generally asked. People generally use may in a formal setting as well, I assume when you are saying casual you mean between friends, meaning there is no formal demand.