I also disagree that shall is 'polite'. It's not. It can be depending on the way you phrase it. For example, if my wife says 'i'm going to do something in the garden...' i can say 'shall i help you?!' but it's pronounced more like 'sh'll i help ya'. Perfectly normal, not formal at all. On the other hand, if i say something like Bill The Butcher in Gangs of New York says - '...shall i keep it (a waistcoat) as a souvenir?!' - that would sound formal - we're more likely to say 'should' instead nowadays.

I can't agree that shall is synonymous with may. I didn't feel like reinventing the wheel, so here is the link to an excellent explanation of the contemporary usage: http://www.dictionary.com/browse/shall?s=t

Yes, it is definitely used. In a course, you are introduced to "shall" at intermediate/pre-intermediate level. You can use "Shall I help you?" but you can also use "may" or (more commonly) "can".

Yep.

I agree with Jerry that 'shall' and 'may' are not interchangeable. The difference is one of meaning, not of formality. If you ask a question with 'May I..?', you are asking if you are allowed to do something, as in 'May I ask a question, please?'. 'Shall' is very rarely used in spoken US English, which is why many Americans, especially younger people, are unsure about its function. Younger Americans come across 'shall' in formal written notices and old-fashioned texts, and presume that it's just a formal alternative to other modal verbs - or a way of making your English sound 'fancy'. In British English, however, 'Shall I ....?' is just a normal way of making an offer. If you're with friends, making sandwiches, and your friend is buttering the bread, you might say 'Shall I slice the cheese?'. As you can see from the context, there's nothing formal at all about it. It's just a way of offering to do something, or of asking if the other person wants you to do this. You can also use 'Shall we...?' to make a suggestion, for example, 'Shall we go for a coffee?'. Again, there's nothing especially polite or fancy about this. It's just a standard, everyday way of making a suggestion - often contracted in natural speech to something that sounds like 'Sh'we....?' - which isn't formal in the slightest! NB You could also use 'Should....?' in all of these cases, and the meaning would be the same. 'Should...?' is more common than 'Shall...?'.