If the box itself is grounded via metal conduit or AC cable, then you just have to run the ground wire from the fixture to a ground screw in the box.

If you want to figure it out, find your power source FIRST. It could be in the ceiling box or the switch box. After that, draw a diagram or look up a diagram online of how the wiring should be for your situation. This allows you to understand what is going on. I get confused with a few wiring schemes (usually 3 ways) and like to look at diagrams to get a grasp of what is going on. Once you have it figured out, then try to get it to work. It is more than likely you have a 3 way switch when dealing with the red which is always hot. That means your power source might be in a different switch box completely. The 3 way would be for the light, and it sounds like a separate hot is there for the fan, which could be capped off and discontinued. You wouldnt be able to use the dimmer for that light if it is indeed a 3 way circuit. 3 way circuits require special dimmers that communicate with each other. You cant dim with one side and expect someone to turn the light on on the other side and have safe passage in the room. The simple way of finding the power source is unwiring the hots in the switches and fixtures involved, leave the hots exposed and not touching anything or each other. Then turn the power back on and take a quality non-contact volt tester and find the live hot.

I think Dave might be onto something with the Horn/Strobe fire alarm ... that would explain wires labeled for speaker and strobe, and data could be to communicate between detectors/alarms and/or call the fire department. DC to me is direct current when dealing with electricity, but I don't know if that's something that would be involved in a fire alarm as a back up or something, and as I said before, I'd be wary of making assumptions on that mess of wires. Ground wires can be insulated, so lack of a bare wire doesn't necessarily mean lack of ground wire, but I certainly don't see any white, neutral wires, as would be seen in a light fixture, and none of these labelled wires should be in a light. Regardless, I think we can conclude that this junction box is not for a light fixture, at least in this configuration. If you want it to be a light fixture, it's going to have to stop being whatever it is and have new wires pulled. Otherwise, I'd be looking for another box that does house light fixture wiring ... and if there isn't one, it may mean that lighting was swagged from a wall or the room was lit without overhead lighting. Changing out an existing light fixture I'd consider; beyond that, I call an electrician. I just remember an electrician we had in to look into a faulty fixture pointing out that whomever had installed it "knew just enough to be dangerous," though fortunately nothing bad had happened beyond some lightbulbs being wasted.

Because the wiring within the cloth covering appears to be modern vinyl insulated wire, you probably will be ok. As I look up at the light over me right now, your proposal is not much different than the 18 gauge zip cord that I see wound up through the chain holding my chandelier, your wire just has an extra cloth covering to hide it. If in doubt, ask the local inspector.