Okay, I’d like to clarify one thing - is there any difference between "nebo" and "anebo" except they sounds differently? Or can "nebo" be used both in yes/no questions and questions requiring a precise answer while "anebo" applies only for the second kind of questions?
"Je živý nebo mrtvý? " - the answer may be either "Je živý" or ""Je mrtvý", I can also ask: "Je živý anebo mrtvý?"
"Je bílý nebo černý?" - depending on the context, the answer may be either 'yes', 'no', "Je bílý" etc. "Je bílý anebo černý?" - the answer may be only 'yes' or 'no'.
Or is it the other way around or is intonation the only way to mark the kind of answer we expect?
Nope. Is she alive... would be JE ŽIVÁ NEBO MRTVÁ. And "it is alive..." would be JE ŽIVÉ NEBO MRTVÉ.
And if it refers to a dog? I suppose in Czech it would be masc. animate, while in English it could be referred to by “it”.
Yes. Dog would be masculine, animate. Mostly because "pes" is a masculine word.