2) A miner could "game the system" by generating a specific char for last position along with the required PoW, though this would require an extra bit of work over non-bad actors.
If you are not generating whatever "random" item publicly, the miners will not know that changing the block header will create any advantage.
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Most of the time when you are generating something random, it needs to be secret. If you are deriving something from block headers, you might be able to generate something random, but it will not be secret.