Is the lightning network usable if you only have the private key to the individual Segwit addresses? Is there a function like sweep from electrum?
You just display the users an invoice which is generated by your Lightning wallet and they pay that invoice. The money will arrive to you in seconds and no confirmations are needed.
I don't understand what you want to sweep. If you want to get the Bitcoin out of your Lightning wallet you can send the funds to an on-chain address of yours through a chain swap service like
https://boltz.exchange/.