Why is exodus offering me a receiving address to receive monero if I haven't done the full sync to its blockchain yet? Are you sure I can't avoid downloading the whole blockchain. There's multiple coins, are all their individual blockchains going to downloaded if I click sync?
Generating new addresses is not an on-chain event, but pure math. You can generate valid addresses while you are completely offline. Basically, you only need to sync up if you want to catch up on the latest blockchain transactions or broadcast new transactions of your own.
Syncing doesn't always mean downloading the whole blockchain! Since Exodus isn't a full-node wallet, it uses external servers to function. But, to keep your info updated, you'll still need to sync it with those servers for the latest transaction details.