Generating new addresses is not an on-chain event, but pure math. You can generate valid addresses while you are completely offline.
So if you just downloaded exodus, never sync before it's ok to generate a receive address and start using it? And after receiving the funds the blockchain ledgers will know I have conducted a transaction even though I never sync before?
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.
The latest blockchain/broadcast transactions are helpful in what situations?
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.
What info specifically of mine?