Think of 在 as a verb. What is the subject that is situated somewhere?

I would say the subject of this sentence is 房间里. And, concerning the subject, it can be said that S 有 O.

If you used 在 the way you did, then the subject in front of the 在 would be missing. So: [Nothing] 在 O

I think you can omit 在 because you've already mentioned the room. It's like saying "There is a chair in the room." , not "There is a chair in at in the room."

Yes. I think they're being stubborn because we're supposed to be learning classifiers in this lesson. Actually, I've never heard 一张桌子 before but I guess it's the technically correct term.