Does GDAX use its own wallet to send Bitcoin on my behalf?
Answers to your questions:

1. GDAX generate a new address for every transaction, they consider a generated address as used. The reason for a new address is to protect your privacy, so a third-party can not view all other transactions associated with your account by looking up an address you have used before.

2. and 3. When a GDAX user sends digital currency from their wallet, the address it comes "from" is one of GDAX's many hot wallet addresses. Any coins sent back to that address would be sent to GDAX, not your own wallet. So yes, kind of like shared wallet.