"František, whose bird is it?"
Translation:Františku, čí je to pták?
"Čí je to pták?" = "Kenen lintu se on?" (Whose bird is it?) - we can also use different (but less common) word orders here: "Čí pták je to?" or "Čí pták to je?"
"Čí je ten pták?" = "Kenen se lintu on?" (Whose is that bird?) - this works in Czech and Finnish, but not in English (it would have to be rephrased as "Who does that bird belong to?" which is a different sentence - "Komu patří ten pták?")
But I'm sure you know this well by know - 11 months after you asked the question :)
It's possible, although somewhat uncommon. Interrogative pronouns (i.e. question words such as who, what, when, whose) normally come first in the sentence. You can stress them by moving them towards the end, but then you also need a suitable accompanying intonation. "To je čí pták?" with the melody peaking on "čí". Or even "Ten pták je čí?!" - this sounds impatient or agitated.