Would your private key exist on both chains?
Yes.
How do you know if you have a wallet that will support both coins in case you wanted to cash out? (For example something like Coinbase.)
Coinbase is not a real wallet, so in their case all you will see is whether they have an interface that allows you to spend the coin on the new chain. You'd have to find announcements by them about what they intend to do in the fork and ideally withdraw to a wallet where you control the private keys.
In a proper wallet, it doesn't matter because you can always export your private keys to another wallet which will give you that interface or you can send BTC to another wallet and then export the private keys to a wallet for the new chain.
How would you access and trade the resulting 'alt' coins?
Depends on whether the chain has replay protection. In BCH's case, you could send away the BTC and export the private keys to a BCH-compatible wallet and then send normally (or if you have a compatible wallet, spend both freely when handled properly).
You should read about
the possibility of SegWit2x replay attacks and keep an eye out for any warnings on bitcoin.org or this forum in case it's unsafe to send coins during the time of the fork.