I've come to the conclusion this is the main sewer drain.

Huh? What do you mean?

The only thing going into this p trap is (I assuming) the AC condensation pipe (the white) and that little copper pipe which I'm assuming has some kind of tap somewhere that you can turn on to fill the trap (because otherwise it'll evaporate and become empty if you don't use your AC) or something else that needs to dump extra water somewhere.

A lot of people have a drain in their basement floor where these things get dumped, but yours is just elevated out of the floor.

It is attached to your main sewer drain (just like all of your other drains) so when you were plunging and causing pressure and whatnot while the main was plugged, it probably backed up and spilled out of this drain.

Greg is right, it's a drain and it backed up with all the plunging and such. You want to be sure water isn't coming back up here. If it is, you still have a sewer restriction.

No there should not be a cap.
If that's the AC (White PVC) it's fine without an air gap. It doesn't connect to potable water. I don't know what the copper is - looks like 1/2" - so not at all clear what that would be. If it goes to potable water (like a water softener, for example) it needs an air gap

As greg suggested, the copper may simply be a water source to fill the trap in the cold months when no AC condensate is available. You need to go to the other side of the wall to check.

I would expect the mainline to have some sort of clean out fitting. If things were backing up out of here you clog was downstream of this fitting. You might want to call a drain service and have them 1) confirm that your line is clean and functioninig properly and 2) explain to you how things are interconnected.

All good advice above so I won't try to improve on that.

Maybe that copper is either a pressure relief line for the water heater or a drain line with a valve that allows the water heater to be emptied. Most water heaters have a pressure relief valve at the top that connects to a pipe that runs down the side. Those are often copper pipes. Usually directed onto the floor if near a floor drain, but this one might have been plumbed to this drain pipe.