1. The "pressure differential" style can go anywhere, but the problem with them is that SOMETHING has to create the pressure drop, and in a residence there are few appliances that will "tax" the water system to cause the initial pressure change. They are also ineffectual in commercial systems when the water piping is large enough to minimize pressure changes.
2. You can create an "air gap" by using a reducing coupling below it as a funnel. The air gap is to prevent "fungus" growing in the line from contaminating the primer device. The hole on its side is a vent to prevent back siphonage.
3. The amount of water discharged is so small, during each cycle, that it is doubtful that there would be enough to prime more than one trap, IF you could even separate the flow into two, or more, streams. For multiple drains you would need a "flow through" or "continuous flow", model which discharges while the faucet it is connected to is being used.
4. Usually the primer is "hung" above the drain with a space between the pipe and the drain, which creates an air gap. When the primer is connected directly to the floor drain, then the air gap is placed below the primer, see answer #2.
5. A check valve would NOT be an effective backflow preventer. You do not need it ahead of the primer, and if installed after it could cause problems. Such as, water would be forced out of the primer's vent holes, and the check valve could inhibit the flow to the drain.