Whose overcoat is this?
Very odd sentence
It's perfectly normal, although it's a bit more common to hear Czyj to płaszcz?
The English is a little weird, though. (But the correct English of "Whose coat is this?" is accepted, so it's not a big problem.)
Does "czyj" change gender with the object being talked about?
Czyj/Czyja/Czyje for singular, Czyi/Czyje for plural.
Just like the Ukrainian "Czy ce pal'to"
Whose is the coat sounds very weird to me as a native English speaker.
'the' is merely accepted, as an... interpretation of 'ten'.
Oh wait, I just assumed that, but it's a starred answer. Removed from starred answers.
To whom does it belong?
Works. If you use "this coat" instead of "it" ;)