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? 
1. No, there will be no new coins when Ethereum 2.0 comes up, it will be a disaster if they do that. And as others have pointed out, a gradual transition to Proof of Stake, much better in terms of: (a) security, (b) scalability, and (c) energy.
2. It will not have any effect on Ethereum Classic.