If I sweep the private key into my Electrum Bitcoin wallet, do the BCH, Bitcoin SV and other airdrops remain attached to the private key?
Your private key is one, but generating the public key from it and broadcasting the transaction in the network requires that the software used supports it. Electrum is a Bitcoin-only wallet, you will need to download the Electron Cash wallet to know BCH and spend from it.
Perform this step after withdrawing all your Bitcoin balance.
I created this guide long ago (2018) but it still works and you can still download coinomi.
Does the CoinOmi wallet keep syncing, I thought it stopped?