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? 
Ethereum 2.0 in layman's terms is simply an upgrade to Ethereum Blockchain. When it says upgrade, it involves transaction speeds, scalability, and many good features that are needed to ease the congestion on the network. I don't know if it can lower the current gas fees we are facing but people are willing to pay the current gas price if in return it will always be a smooth transaction.
There is no new coin and it doesn't have to do with its market price.