Obviously every hospital is different. In mine I welcome birth partners into theatre whether you're having a spinal or a general anaesthetic. The obs anaesthetists association debated this one about 10 years ago and the majority view was that the father should be allowed in (with the proviso that if he were to be asked to leave, he went without arguing).
If my DH were to be undergoing major surgery, I'd be worried from the moment he went in to theatre to the moment I was allowed to be with him in recovery. Any support offered from family members would be hugely appreciated. For your DH, this is his first experience of childbirth and so he is going to be worried about both you and your baby, I can understand him wanting to have some support there himself while you're asleep.
They can keep the baby in theatre with you while you're asleep, the midwife just has to stay in there to look after it while the operation continues. In practice most fathers want to see their child as soon as possible which is why the baby is allowed out to be with them until you wake up.
In most places once you are awake you will be allowed one birth partner with you and the baby in recovery.
Don't be surprised if you are prepped, draped, catheterised etc while still awake. Many of us follow that pattern and don't give the anaesthetic drugs until the very last minute (surgeon standing there poised with blade) to minimise the time you spend asleep (during which time the baby is also getting some of the anaesthetic) before delivery. Elective GA caesars I always wait for the catheter, painting and draping (none of which are painful, just a bit cold) before I get the patient off to sleep. The surgeon then waits until I give the go ahead that the airway is safe before they start cutting.
You can't have a "lighter" anaeshetic. You are either awake or asleep and will be given enough anaesthetic to make sure you are asleep. Some people wake up faster than others. Sometimes they use monitoring to check how asleep you are, sometimes not - I tend to, but I use it for most things.
You're not being selfish to want to be the first person to see / cuddle your baby. But your DH isn't being selfish to want some support when the woman he loves undergoes major surgery to have his child. I don't know if compromises like "can she wait until baby comes out and then go away" would work - for one, would she be able to go away knowing her son was holding his precious first born, without a quick sneak peek / cuddle? Would your DH want her to go and leave him literally holding the baby or would he be too nervous? I think this is just one you'll have to talk out. Be honest with your DH and your (f)MiL about how you're feeling, don't suppress any jealousy or resentment or upset, because if you do it'll come back to bite you later. See if with honest talking, you can come up with a solution that allows your DH some support and you some reassurance.
You'd all hate where I work. Caesar theatre has a family viewing room with large window - parents, siblings, aunts, uncles, friends, you name it we've had them in the whanau room, with one family member in theatre. Definitely a spectator sport / family event over here. I think the most we've had in one delivery room is about 50.