In this case I think I would translate this as "Who has the pear?" because, although "ta hruška" means "the pear, that pear" being both interchangeable, in this case it doesn't makes too much sense to ask for "that pear", because it sounds like your pointing to "that pear" placed there. Is it right? Or I missed something?
Yes, the demonstrative changes based on gender, number, and case. In the nominative case, it would be ta hruška if hruška were the subject of the sentence. But since hruška is the direct object in this sentence, we need the accusative case... and that gives us tu hrušku.
Please see my reply to BriannaTur103748. It describes the change in the noun as well as the change in the adjective. Also, VladaFu's reply to the same question suggests reading the Tips and Notes, which are very helpful. Always read the existing discussion before posting.