Camcorder + capture card setup is the way to go for the playfield. C922s have been found to be buggy and inconsistent when using its 60fps mode. At this point I suppose you are stuck with it, but anyone else trying to get into this, do it right and avoid these things as playfield cams. You will be happy you did.
For the other cameras you want different models. Multiples of the same make/model can cause device conflicts as well as confuse the software. This can make the configuration process tedious and difficult. I use a C920 and a 920-C for the player and the DMD cam (the later is the 920 business model and doesn't conflict with the consumer model). If you need a third one, get another unique model. Honestly though a fourth cam doesn't add much to the stream and I don't feel it's worth the trouble setting up. At the very least, ensure you have three stable cams (playfield, dmd and player) before you decide to go all-out on a fourth. Or even better, make sure you have two fully consistent, stable cams before you rig up the third.
On a laptop setup like that you will need a good, powered USB hub as well if you really have to have four cams. Otherwise that MSI system linked to should be fine out of the box as-is. It is current hardware. I am using a laptop that's got an i7 generations older and am not running into the issues you mentioned. You shouldn't need to drop the other cams down to lower resolutions either as your system is plenty beefy enough as is.
Something else of note, when using Logitech's cams, you want to disable the added processing for lighting ("Right Light", low light compensation, etc). You will find the motion in the camera has a choppy, laggy nature to it when these things are enabled. Things will also get choppy if you have your exposure and gain set too high, so make sure these are lower. Have lots of light in your room to brighten up the room without relying on the webcam's built in post-processing.