grammaticality - Is it incorrect to begin a question with some qualifying phrase or statement?

A few example questions would be:

If you're not too busy, can you wash the dishes when you're done?

I don't think I can; can you wash them?

I've done them too many times this week, don't you think?

The first example has a (preposition?) that can't stand on its own as a complete sentence, and the last example adds a question that would be a fragment on its own, whereas the second example could be rephrased as two separate sentences and read mostly the same. The crux of my question is, if I may make an example of it in itself, are all or any of these sentence structures right or wrong, and should I be rephrasing them as separate sentences? Also, right or wrong, can you provide the terms for those parts of the questions and sentences so that I may better reference them in the future?

Answer

Your first example is a common structure for a polite request structure, beginning with "If you're not too busy...". The word could might make it more polite than can, especially with a "please".

The second example feels a bit incorrect without context. I would use two sentences, or something like:

"I don't think I'll have time to wash the dishes before I have to go. Could you please wash them, if you have time?"

But to make something that feels correct with the semi-colon:

I don't think I can get to the dishes; could you wash them?

The third example is called a "tag question" and is perfectly correct. You may have noticed tag questions are used a lot by British speakers of English, right? ;-)

Wikipedia actually has a pretty good explanation of tag questions, but they tend to be separated by a comma at the end of a statement, like:

[…], isn't he?

[…], weren't you?

[…], aren't they?

[…], don't you think?

[…], isn't that right?