Simply speaking, "czy" is in questions, "lub" is in other sentences. There are exceptions, but that's probably all you need to know for now.

"Filiżanka" is only appropriate for a teacup or a coffecup (the little ones you can made out of porcelain). All other cups are "kubek". A mug also can be "kubek", but if it's a larger one for alcoholic beverages (like beer), we would say "kufel".