is second 'a' like in kağit? imkânsız?
Yes, although it seems it's not written like that much anymore.
why isn't "that" acceptable in this case?
Can this also mean 'why is he/she impossible?'
"Impossible" when used with people means something like "hard headed." This meaning doesn't exist in Turkish :)
Is 'imkanli' usable in the sense of 'possible'?
No.we use "mümkün" for that,and we rarely use it as an adjective,to say something like "possible questions" we use "olası" or "muhtemel"
Why it is rejected: Why it is not possible
This is incorrect in English. In English, you reverse the order of words in a question: Statement: It is possible. Questions: Is it possible? Why is it not possible? (we often say Why isn't it possible?)
Why is this impossible?
I marked wrong
Because of "this" (bu) vs. "that" or "it" (o), I would say.