Shall and will are different, and you need to know how to use both.  If you are using textbooks in a course, you'll be taught the difference around pre-intermediate level.  Yes, this means it is important.

We typically use shall for offers and suggestions in question form, and when you use I and we.

Shall I open the window?

Shall we go out for dinner?

It is not possible to use will and keep the same meaning.  "Will I open the window?" means you don't really know!  So, you can start practising your Shall I...? and Shall we...? phrases right now. :)

If you get to upper-intermediate level, you'll see how shall is used for serious obligations. The word shall is often found in legal contracts.

There's no difference between US and UK English here. It's a standard usage.

Here's a handy guide for these modals... it's not a proper definition, but it gives you the idea behind each modal:

will -> volition

shall -> obligation

Must has the sense of obligation/motivation, but the origin is a little more difficult to describe. Anyway, knowing how shall works can explain the different uses of should (I mean, its uses aside from advice).