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? 
It won't be a separate coin. ETH 2.0 just means that it switches from proof-of-work to proof-of-stake and as others already mentioned phase 0 of ETH 2.0 is already ongoing.
So it's definitely not a reason for concern but, especially in this times with the extreme high gas fees withing the ETH-Network, a thinkg to look forward to as it should increase the scaleability of ETH a lot and therefore also bring the transaction fees down.