Is it really wrong to say "Proč moje košile nosíš?"?
Yes, it is really strange. Like if you were wondering why he is wearing them instead of doing something else you expected more he/she would be doing with them.
But is it still grammatically correct?
Košile is singular. Is it also plural? I assume it’s in the accusative case, right?
Here it is plural accusative. Plural nominative and singular nominative is the same. Also singular genitive. It is the "růže" inflection paradigm https://en.wiktionary.org/wiki/r%C5%AF%C5%BEe
Why do you wear my shirt(S)? Is this Proč nosíš mojí košili , did I do right Vladimir? Is this casa we wouldn't use "mou" isn't it?
Your sentence is almost correct for singular (one shirt), but the i in moji should be short. Moji and mou are then equivalent and equally possible. See http://prirucka.ujc.cas.cz/?id=m%C3%A1
Alright Vlada. Thank you for the help!