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 phase 0 has already kicked off on December 1st. Ethereum 2.0 phase 0 does not yet allow the creation of wallets on the Ethereum 2.0 blockchain. This feature will be added in Phase 1.Eth2.0 = Eth1.0. 2 parallel blockchains will work on different POS and POW algorithms.
https://launchpad.ethereum.org/
"What is eth2 phase 0?
Transitioning from proof-of-work (PoW) to proof-of-stake (PoS), eth2 will be launched in at least 4 phases. The idea is to break up this transition into phases each focusing on a different aspect of eth2.
Phase 0 contains all of the machinery behind eth2's PoS consensus, it tracks the validators and their balances.
Phase 1 handles adding, storing, and retrieving the data associated with eth2's shards.
Phase 1.5 updates Ethereum as we know it today from PoW to PoS by making it a shard under eth2.
Phase 2 Phase 2 adds execution to the remaining eth2 shards which enables smart contracts to run on all of the shards."
