We installed a hood fan/microwave combo at our last house. It came with steel brackets that were to be installed on the back wall and on the cabinets alongside the microwave and then it slid in and you put in a couple of screws to secure it
The duct for the hood fan went up through the cabinet above and then we ran it outside. We had to cut holes through the cabinet to fit the duct, and then we used spare MDF to box it in so that it wouldn’t show and so that it would be quieter.
We somehow found a microwave/hood fan combo that went to 600cfm and was super effective. It also worked really well as a microwave too. Of course it was from Sears/Kenmore circa one year before it all shut down in Canada.
Over the stove microwaves have a vent system. The one we bought for our old house could vent out the back or up on top, depending on how the exhaust vent is structured. There was also an option not to vent—just to recirculate basically—assuming there was another venting system.
I can’t exactly remember how it was installed but I think it was screwed into the cabinet on top and then there was a wall bracket thing that it kind of locked into/sat on the edge of.
I don’t think ours vents (or at least not properly). It has a cabinet above it with no area to vent.
Anyway, I have forbid my H from turning on the vent because the last time he did it left a an oily residue on the ceiling. I think it’s just venting up and taking greasy air toward the ceiling? I don’t know. I told him to knock it off.
There are some that don't vent outside. My sister's has slats that open in the front at the top and it just recirculates the air back into the room, which is super fun for my 6'4" BIL, LOL. My experience has also been that most don't sufficiently hang over the front burners so it'd have to be strong enough to pull air at like a 45-degree angle, which isn't really a thing with microwave vents.
I hate over the stove microwaves because they're useless if you need actual venting.