Is "Can I have a cup of coffee?" polite?

To Sam's comment, the more pedantically-minded English speakers will state that beginning a question with Can I is to question one's own ability rather than to make a request. To satisfy those people, you're better off using may:

May I have a cup of coffee?

If you want to be polite about it, add a please:

May I have a cup of coffee, please?

If you want to be especially considerate, like when you need to get some coffee from a person who's had a bad day, make the hassle of getting the coffee the focus of the request instead of the coffee itself:

Could I trouble you for a cup of coffee?

Finally, if you're familiar with the person, you can just skip the question altogether without offense:

Pass the coffee, buddy.

It's common to assume a person who's looking for coffee is suffering from caffeine withdrawal, allowing for a good deal of leniency when considering how polite the request for coffee is.

Can has an abruptness to it that could come off sounding demanding.

Could I have a cup of coffee

or

could you help me find the beans.

would be better, and it never hurts to say please:

Could I have a cup of coffee, please? What do you mean this is a sporting goods store?

This is a slightly tongue-in-cheek BE vs AE thing.

American English - Would you bring me a cup of coffee? (trying to be polite)

British English - Could you bring me a cup of coffee? (I assume that you would - so it's a simple question of do you have the capability to make coffee)

This may be the only example in the language where the AE form is more deferential than the BE!

Is "Can I have a cup of coffee?" polite? What if I tell "Can I have something?" to a seller in a shop?

Yes, it is polite. We all use the modal verbs for polite requests. 'can' is not as polite as 'could' or 'would' but it's commonly used in situations like this. 'will' is even possible. The nuances of each can depend upon a number of factors; familiarity, the degree of deference one wants to express, ... .

The notion that 'can' means "ability" in these polite request situations is the result of an old, poorly thought out prescription. Corpus studies show that 'may' is rarely used adult to adult, compared to 'can/could' for these types of polite requests.

The meaning of most polite requests using 'can/could' is "is it possible ...". It has little to nothing to do with capability or ability for a rational person wouldn't consider whether a cafe/restaurant has the ability to make someone a cup of coffee.