Tell me please which question is correct?
Is there anybody sure that he is right in the room?
Is anybody sure that he is right in the room?
I feel that both sentences are correct, but I cannot see the difference. Is there any?
Your first sentence is just about grammatical, but it is not idiomatic, and does not have quite the same meaning.
Is there an X?
can usually be replaced by
Does an X exist?
So "Is there anybody sure that ... " means "Does there exist somebody who is sure that ... ". Formally, this might mean the same as "Is anybody sure that ... ", but pragmatically it certainly doesn't.