I wonder if they could add a trash launcher to the ISS. If some kind of spring or slingshot threw trash retrograde with sufficient velocity, there should be two advantages:

1) The trash would be put in an orbit with a lower periapsis, increasing atmospheric drag causing it to re-enter and burn up more quickly.

2) The orbit of the ISS would be boosted by dV of the trash, reducing the fuel required for the next boost.

Would you have to launch half the trash at apoapsis and half at periapsis to keep the orbit from deviating? (I've only played ksp.)

Since to not alter your orbit you'd need one prograde moment and one retrograde, one of the two launches you describe would raise the apogee instead of lowering the perigee, defeating the purpose

Just a question: Would pushing the trash away also push the space station in the opposite direction?

Yes, but that is part of the point. Atmospheric drag causes the orbit of the ISS to continuously decay (fall deeper into the atmosphere).

The ISS must expend fuel to push its orbit back up periodically. The momentum of the trash would help push the ISS higher.

yep, equal and opposite reaction. they'd have to counteract it.

Anyone know why they didn’t just keep it around, surely having a few old batteries and a bit more shielding around the ISS might have come in handy? Was it risky to keep them around or did the extra weight (even in a basically weightless environment) come at a cost to maintaining orbit?

The ISS has to regularly perform station keeping, i.e. it has to expand fuel to raise its continuously decaying orbit. Any additional weight would increase the fuel required for this.

why not push it out of orbit, away from earth?

I'm likely wrong as my only experience here is playing ksp, but I would think at no point in the ISS orbit would it be able to economically fire trash at an escape velocity, and I wonder if doing so would disrupt the ISS orbit. (depending on the weight of the trash)

That means it will stay around longer, thus will be a hazard longer. Allowing things to de-orbit and burn up on re-entry is a safety feature.