But is it safe to send BCH to Coinex from my Keepkey wallet if I still hold my BTC there?
If your Bitcoin and Bitcoin Cash are split, sending one of them doesn't influence the other. You can send your BCH without sending your BTC.
However: BCH, BSV and BCH-A don't have replay protection. That means that sending one of them will most likely make the coins move on the other chains too.
I understand that Coinex will automatically split BCH into BCH, BCH SV and BCHA if I send my BCH there.
I don't know if this is true, I've never used Coinex.
What happens if I send an unsplit coin to a wallet that supports that coin but not its airdropped "child?"
In general: you lose the Forkcoins! Most exchanges don't recover coins sent to the wrong chain.
That's why it's better to send Forkcoins to wallets under your control until you've successfully split the coins.