No they're not. There's no air for them to breath.
If something logically cannot exist, then there's no doubt. The idea of any sort of Supreme Being, creator or not, still around or not, isn't logically impossible; it's improbable. The wises of Sages have frequently repeated that God cannot be known by intellect, and many have still said that Scriptures don't paint actual portraits of God, but rather just how he/she/its been perceived.
Do unicorns need air? They are magical... They can use magic to not have to breath at all.
It depends, it's just the counter-effect of its omnipotence allowing it to exist (depending on which kind of God). According to science, though, it is illogical by a lot.