In other words, if I double my number of video cards, should I expect twice as many coins mined per unit time? From what I can gather, coins are rewarded in a winner-take-all fashion, so if you have more computing power, your pool can solve the next block in the chain before anyone else, and so rewards would favor people with more computer power in a non-linear way.
Well, it's close to linear. It basically is. If there are 200 billion hashes per hour on the network currently, and if you are providing 200 thousand of those hashes, then you have a 1-in-a-million chance of scoring a block, right? If you double your hashing from adding a card, then you have a 2-in-a-million chance. But that's using rounded numbers (which are most applicable to the real world generally).
But let's imagine that you decide you want to run a million of those 200K hash cards. You'd be performing 200B hashes per hour. But the network was already at 200B/h, so now it's at 400B hashes per hour. So instead of you having a million-in-a-million (100%) chance, you'd only have a 50% chance of scoring any given block.
To address the added concern, the law of averages effectively says that only in appearance will the pools take greater than their share of winnings. If a pool is mining at 40% of the hash rate of the network, they have a 40% chance of winning a block. It really is as simple as that.