If Ethereum 2.0 comes out, will that be a new separate coin and will it bring down the value of Ethereum and Ethereum Classic that is currently in Coinbase or Gemini? 
There will be no fork coin or separate coin from ethereum. Ethereum POW will be only converted into the POS coin. The POS coin of ethereum is alive now. Fork coin gives a very bad reputation to the platform and that's why a lot of people didn't even wanna see that to happen. Nothing happens with ETC in the future.