Once a paper wallet private key has been sweeped, can Bitcoin still be safely sent and stored to the address of the sweeped private key?
There's risk the private key was exposed when you sweep the private key with device that connected to internet. It's safer to create new paper wallet.
I would rather generate multiple paper wallets in advance than generate them on an ad-hoc basis. Generating paper wallets in batches means that you are never having to consider compromising on security or best practices because you are in need of a new paper wallet.
I would advise against using paper wallets though. It would be better to store your private keys in a USB stick or computer that never touches the internet. When you are ready to spend (some of) your bitcoin, you can sign the transaction offline.