To exchange themselves they only need to load their existing wallets in an existing BCC client.
But is that really what's going on? As I understand it, each BTC also became a BCC at the time of the fork, and owners can now decide on which network they want to use these units (CB users are not technically owners though). No free BCCs were distributed.
If I were to decide to move 1 BTC to a BCC wallet now, I would buy something worth $650 for something worth $2700.
That is my very tentative understanding at least.
The chains do not talk to each other in any way, so you can spend your BTCs and BCCs independently. So yes, free BCCs were given to everybody with BTCs.
Imagine it as if someone copied the entire history of bitcoin and just renamed it and started running on different nodes. That's basically what happened. If the bitcoin history showed you owned 10 BTC, then the renamed history shows you own 10 BCC.
No. Each BTC also became a BCC at the time of the fork, and now exists independently on both networks. Owners are free to use them on each network. That is the concept of a fork -- it's an intended double spending attack. There is no such concept as "moving 1 BTC to a BCC wallet". Moving BTC means a transaction on the BTC blockchain. Moving BCC means a transaction on the BCC blockchain.
Since BCC doesn't have enough hash power to mine blocks trading/withdrawal would not work anyway.
Coinbase made a colossal mistake in outright ignoring BCC and will likely get sued (and lose) because of it. If I had 10 BTC on CB and missed out on the opportunity to trade or sell $6,500 worth of free BCC I just got, I'd be mad too.
They need to act quickly before it goes to court. Someone else commented that they don't even need to support trading BCC, all they have to do is provide a way for customers to withdraw their BCC, and they're in the clear.