Can I activate engines on spent stages?

To avoid cluttering my sky with bits of engine and fuel tank, I had the thought to strap some solid fuel boosters to stages pointed retrograde and activate them during staging in order to deorbit them. However, on the couple rockets I've tried this end up being damaged by their thrust and/or the brief shock when decoupling.

Is there a way to separate the stage then take control of it later to chuck it back into oblivion? Alternatively, can I have it just fire after a set time?

Or is there a better way to deorbit stages?

KSP does not allow remote control of unattached pieces. Anything not attached to your main cockpit automatically becomes directly uncontrollable as soon as it detaches.

That said, there are a few options available, if you're looking to de-orbit spent stages.

1. Create a dedicated deorbit vehicle that catches your debris, and burns retro to shed some speed. That's rather difficult, as everything needs to be intercepted individually.
2. Attach a probe cockpit (or MechJeb component) to the stage. This will allow you to switch your focus to it. I'd recommend ensuring your primary stage is in a stable orbit before doing so, as fully deorbiting something will take quite some time.
3. The last option is to turn off all debris entirely, as detailed in my answer here. That's the easy way out, but without specific tools to deorbit junk, it's rather involved to manually clean it up.

Any way you look at it, currently KSP doesn't allow control of parts that are not directly connected to your current focus.

Yes there is, if you have a component capable of issuing commands, and it is properly manned/powered. This typically means attaching probe bodies to each stage that you will want to control later.

I find this sub-optimal at best though. Performing a post separation deorbit typically means you need to save some fuel in order to do those burns. And since probe bodies don't have a lot of torque, you're going to have to make sure that you have sufficient RCS fuel and thrusters. There is one advantage here, in that RCS is typically enough by itself to deorbit something. However, this all adds weight to your rocket, weight that's better spent as saved fuel in sending your payload where you want it to go. For typical real life missions, hoisting an extra pound of payload to orbit could mean burning an extra five ten pounds of fuel.* That can get very expensive, very fast.

In my opinion, you're better off setting up stuff such that it falls back to the ground without your intervention, or setting up trajectories that will cause spent stages to impact the Mun (or other celestial body).

_{* This page lists some mass fractions required to achieve SST-LEO, as well as typical mass fractions for existing space vehicles. In an ideal world, a 5:1 fuel to payload ratio is theoretically possible, but that ignores the realities of atmospheric drag, incomplete combustion, and a number of other factors. The ratio for Soyuz is almost exactly 10:1.}