What is your hair color?
This is perfectly understandable, and I'd know what you meant. But I think it is more likely that we say instead:
What color is your hair?
Then we have your other suggestion:s
What hair color do you have?
What color hair do you have?
To me, using hair color rather than color hair implies that the person has dyed their hair, and you're asking about the product they used to dye it. What color hair do you have would make sense to me as simply asking what shade their hair currently is (no reference to dyes) but I still think What color is your hair is much more common.
For shoes, either of these are understandable:
What size shoe do you wear?
What size shoe are you?
What size are your shoes?
But I think the most common expression would be:
What's your shoe size?
I don't quite understand your tool shape comment (what would be possible answers to the question "What tool shape do you make"? Then maybe I can edit to elaborate). But the word order depends on exactly how you're asking, in any of these cases. In some cases you're asking what the color (of their) hair is, in some cases what (is their) hair color. Hair color is a single object; color of their hair is talking about their hair and then asking about the hair's color property. So it just depends on how you structure the sentence.