"Do we have a shirt?"
So you don't have to say "Czy mamy," just "mamy"... Is that because it's in writing and you can see the question mark, or would tone imply the same thing?
You can omit 'czy', but then you have to make it obvious that it's a question. For learners, it really is a better idea to use it.
Am I wearing clothes ? or am I currently naked ? I'm not sure, I should have a look...
Why doesn’t koszula make a Genitive? I thought all objects which were Accusative in a statement, will transform to Genitives in a question.
No, not in a question. In a negated sentence.
And not 'all objects'. If the verb takes Accusative, it gets Genitive instead when negated. Only in this situation (which is quite common as Accusative is a common case). But other cases stay the same.