Well, yes, it is sketchy wiring! That visible electrical box should be flush mounted in the plane of the drywall.

That incompetent installation violates code and is a potential hazard. Remove that existing box and properly install a new one that is mounted flush with the ceiling. After that, I suggest you have an electrician inspect all your wiring. When you have a home with crappy DIY work like that, it's often not the only instance of bad or potentially dangerous work.

So it looks like there's more than just the two wires meeting in this box. Should I just run the other two wires that feed power to and from my light to a flush box, and leave this one and the other two wires not connected to my light there?

No. Code requires that all wire connections (i.e., wire nuts) must be in boxes that are accessible. If you just put in a flush mounted box it will effectively hide that illegal box, which causes problems when electricians are looking for future electrical problems and there are hidden boxes. Do this right: undo the connections in that box, remove the box, and put in a new, flush-mounted box where you rejoin those connections and also have the connections for the lamp. Also, make sure the light fixture is mounted to the box directly or with a fixture strap rather than being held to the ceiling by a couple of screws into the sheetrock. If you're not sure you understand all of this or have the skill set to mount the box correctly and make proper connections, please get a knowledgeable friend or an electrician to help you.