(1) "Whose shirt is that" is correct. (2)" Whose is this shirt" is also technically correct, but it sounds odd enough in modern American usage that my urge is to find a noun after "whose" to complete the sentence.(Note: You can definitely say "whose is better?" for "whose pie is better?") Grammatically, "whose" is both a possessive adjective (explaining its use in (1) above) and a possessive pronoun (explaining its use in (2)). Aside from the different use of "this" and "that" in your sentences, both mean the same thing. The asker desires to know "to whom belong (this/that) shirt." However, I would definitely recommend saying (1) to sound more natural in everyday speech.

"Whose shirt is this?" is more natural, but yours is also correct. "Whose is this?" is a common way of asking the question without using an object like 'shirt'. The first sentence is perfect.

Yes both of them are correct. The difference lies in the meaning. 'This' is used for things which are near or close to you but 'that' is used for things which are away from you. If you are holding the shirt in your hand you could use ' whose shirt is this' but if you are pointing towards a shirt kept at a certain distance, you could use ' whose shirt is that' Hope I have answered the question. Thank you.