"What color shirt does he have?"
Translation:Minkävärinen paita hänellä on?
The English sentence is unnatural, but I'm not sure if it can actually be rephrased well without also having to modify the original sentence.
Something more natural could be, perhaps, "What is the color of his shirt?" or "What color is the shirt that he has?". The "does he have" construct only really works if you ask "What kind of shirt does he have".
Partly so. If it were the partitive, the person would only own a part of that shirt, not the whole shirt. So, if you have an uncountable noun, let's say "nauha" (string), you could ask "Minkä väristä nauhaa sinulla on?", assuming that that person has some string but not knowing the colour of it. If you were talking about a specific piece of string and referring to it as "nauha" ("nauha" as a countable noun), then you'd ask "Minkä värinen nauha sinulla on?".