1. Can I keep using the old wallet now until I install the new one?
2. I do need to sync to the network (I've been afk for a week) before transferring the coins to an exchange, right?
3. Should I just send the coins to an exchange, then install the new wallet, then transfer from the exchange to that wallet?

I'm using both wallets till I can see it working 100%. So to answer your question you can use both on same computer with no issue.
Old wallet is located in C>Program Files>BitBay
New wallet is located in C>BitBay