yes, you can skip "czy" in questions, and you can always skip "ty".

Keep in mind that using "czy" makes a question more clear, especially if you are not a native speaker and have trouble with pronounciation.

They are both Instrumental forms, so let's rather discuss the Nominative, basic forms.

The difference between "chłopiec" and "chłopak" is a matter of age. "chłopiec" is a younger boy, "chłopak" is older. But there's no set border, it's a subjective thing. Personally I'd say that "chłopiec" should 'end' not later than 12-13. Rather earlier.

Also, "chłopak" means "boyfriend", if it's "someone's chłopak".