It means that. It cannot be that. One cannot be thirst, one can only be thirsty. thirsty is žíznivý, but that is not too common and would mean very strong thirst.

I believe the standard constructions for being thirsty or hungry in Czech are "mít zizeň" and "mít hlad," and that's just something we need to adjust to, coming from English. But you're right, translated literally, they sound very odd!

Kdo is third person singular. Because it is the subject of the sentence, the verb is also third person singular.