The electricity will be reduced by the fact that Round 1 is not computing-intensive
Regardless of the complexity, if I use more power to increase my hash rate, then I will have a better chance of getting to the second round. Right? So, why wouldn't I use as much power as I can to get into the second round?
You could but if I'm understanding the OP proposal correctly is that he will only allow a certain # of miners into round 2, such as 100. How those would get chosen he said it is based on the first 100 that solve the puzzle. So in that sense, if you increased your hash rate then it would increase your chances of getting into round 2. So your question is valid. The answer is you would use as much power as is necessary to be in the top 100 but no more.
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I think a solution would be to eliminate the PoW in the second round and simply make it a random selection (somehow). That would cut the expected block reward by a factor of N2, which would cut the power usage in the first round by that same factor.
Not sure on that. But why not just have a single round 1 and just select the winner at random without eliminating anyone? Why wouldn't that work?