I have considered the Q question from time to time, and a possible motive for his actions has occurred to me. I think Q–the DeLancie Q, that is–is trying to save his people.
It’s been established that the Continuum has, from its own perspective, rung all the changes. They have exhausted the novelty of their power and knowledge, and the endless monotony is wearing them down. There has already been one suicide. It’s only a matter of time (albeit probably a very long time by the standards of most races) before there are more.
DeLancie-Q has decided that the Continuum needs a kick in the pants, in the form of new peers. Friends, rivals, enemies, or all of the above, as long as they have a new perspective. He’s playing a long con, duping both the Continuum and his targets into thinking he’s just meddling out of boredom, but his actual aim is to accelerate and guide the development of a lesser race (or races–he could be working on hundreds of others in parallel, after all) into a state from which they can interact with the Continuum on their own level.
He can’t just wait for events to take their course; there’s too great a chance that the Continuum will collapse before peers arise naturally, especially if his most promising candidates manage to wipe themselves or each other out. He also can’t just fix things for them, because they won’t learn anything. He can’t even guide them too much, because if he did, by the time they were capable of rivaling the Continuum…they’d essentially be the Continuum.
So he tricks and teases and threatens his favorites. He exposes them to new things. He makes of himself an adversary to challenge them. He pisses them off, so they don’t want to be anything like him, and tempts them with tastes of the power he wields, so they will want to match it. He even warns them of danger, though he doesn’t actually protect them. He’s teaching them.
If Q is a god…he’s Coyote.