Is essay typed or written in offline GRE exam ?
Is there space for rough writing during exam?
It depends on which offline test you’re taking.
There’s the paper-based test which is like the SATs, you need to fill in bubbles to mark your MCQ answers and handwrite the essay.
But I’m pretty sure that ETS sticks to taking computer-based tests - where you’re able to type and click unlike the former. This mode is technically online, but you can give the computer-based test at home or at a GRE test center.
The other reason the GRE is generally computer based is so that the section difficulty presented to you is adaptive to how you perform in real time as you submit each section. In case this sentence doesn’t make sense, I suggest you refer to Greg’s videos from Day 0 of the study plan.
As for rough writing - you’re supposed to use an A4 dry-erase board (a small whiteboard) for rough work.