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? 
No more fork coin from ethereum blockchain and you will never see that happen again. it's clearly stated by ethereum core developers in the past if ethereum classic will be the first and last fork coin caused by the DAO. You will not get a new coin dude. Ethereum core developers already anticipated it to happen again.