Would your private key exist on both chains?
if there will be two chains then yes. since the chains need to be backward compatible, the private keys and all the transactions up to the point of the fork will be valid on both chains.
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.)
How would you access and trade the resulting 'alt' coins?
something like coinbase is not a wallet exactly. and you need to ask them, but they usually make a clear statement when things are close and clearer. just like they did back in July (before the August fork).
if you have your private keys you have your coins. if you don't see a private key (like in Coinbase) you own nothing but a promise.
as for trading you should see which exchanges list them. for example BCH was listed on bittrex from early days. other unknown exchanges like ViaBTC also had it but they are never trusted enough.