Electrical – Should ceiling fan mounting screws thread into the box or the framing behind

Ceiling fans, when properly mounted, are put on specialized mounting/junction boxes that are especially strong, have a very secure mount for the fan and are hard screwed into a beam or stud.

What you have there looks like a pancake style fan box. See this example:

Every junction box has threaded holes for mounting fixtues such as your track lighting base. The studs in your picture appear to be the bolt holders for the fan. If they are stripped, it is not safe to hang things from them and oversized bolts are a risky work-around.

It may be that other screw holes in the bottom are properly threaded for standard fixture mounting. Check them using the bolts that came with the fixture. If they are not (and there are usually several unthreaded holes in junction boxes), you should probably replace the box with a new one with intact mounting tabs/holes.

The bare copper wire and any green insulated wires are meant to be connected together, making it a "party of three" as you stated in your question. That said there is another consideration to take into account as well. These safety ground wires are also meant to be connected to a common ground point of the electrical box as well.

If one of these wires (green or bare copper) coming out of your box already attaches to a safety ground point in the box then you have no more to do than the "party of three" connections.

If neither of these two wires are connected to the electrical box safety ground then it will be necessary to add an additional pigtail green or bare copper wire to the box safety ground and join it with the other three making a "party of four" connection.