I’ll use Deathrain piloting a TIE Punisher for my example, but any other examples anybody can think of please include them.

Deathrain’s ability states that “After you drop or launch a device, you may perform an action”

Is this a free action or does it take the place of my action that round?

I am of the mind to say that since it does not explicitly say “You may not perform another action this round” that it is in fact a free action.

You're correct.

The 'free action' terminology was removed from 1e, presumably because it caused a tonne of confusion, which means that unless something specifically calls itself out as using your perform action step or blocking your actions later in the round, it's what would have been in 1e a free action.

Obviously you cannot repeat actions, so if you use your Action to Focus, you cannot Focus as your action later.