If miners shut down, mining rate slows, supply goes down, so price should go up. Right?
Bitcoin algorithm automatically adjusts every 2016 blocks, to make the average block time 10 minutes. If the number of miners drops, the algorithm will adjust the difficulty of finding a 'good' hash. Meaning as an individual miner you have a better chance of finding the correct hash, but the overall network has the same rate.
If Bitcoin transaction rate would be significantly reduced for weeks or months, would that kill it in the sense that Litecoin or Ethereum would take over?
The transaction rate would only be affected for a max of two weeks as I understand it. As for how the 'market' would react if all Chinese miners dropped out, I have no idea :)