Yes, broadcasting solved blocks quickly matters, but the principal deciding factor in how quickly you find blocks, for any given difficulty target, is how quickly you compute hashes, independent of how quickly everyone else is computing hashes.
Yes, you will mine blocks at the same rate you were before the network hash doubling, until the difficulty reset.
However, this is still zero sum and the network just beat you. You will mine less blocks now that the network increased even prior to the difficulty reset.
The reason why is the difficulty algorithm is block-based not time-based. After N number of blocks are mined, difficulty will then reset itself. You only get that number of blocks to mine at a given difficulty, not a set window of time. Once those blocks are solved, it's on to the next difficulty window regardless of if it took a day or 10. Someone doubling the hashrate the moment difficulty resets will cut your expected return in half for that round as "people other than you" are now solving blocks at double the previous rate.
Yes, but the difficulty adjustment will now also happen more quickly because the confirmation times will decrease due to the higher hashing power, and then after that you'll be finding blocks at a much slower rate.
The idea of an "easy cycle" and "hard cycle" is interesting, but I'm not convinced it's an issue, because if many miners decide to only mine during the easy cycle and not at all during the hard cycle, the difficulty adjustment will mean that the hard cycle will take much longer (due to increased difficulty AND decreasing hashing power) than the easy cycle, in a way that evens out and removes the arbitrage of only mining during the "easy" cycle.
apart from a very small percentage of the time where you find blocks at roughly the same time as someone else
Not only that, every time a block is mined you have to spend time to construct a new block, which takes time.
This is precisely what it is, every pool operator I've worked with put considerable engineering expense into ensuring that they could broadcast their solved blocks as quickly as possible to the network.
This is the definition of a race. It's somewhat common for a pool to mine a block a second or two before another pool.
Mining is zero sum. If you have more guesses per second than your neighbor, you will win. Only in a pedantic sense could this not be considered a race. If my neighbor then doubles his hashrate, I will lose comparably to his increase. We just each make less overall money as the difficulty increases since we're engaged in an arms race with each other.