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.
And therefore can that priavte key be sweeped again in future?
Yes, as long as the address from that private key contain Bitcoin.
Or after a private key has been sweeped does the address become invalid?
The address will remain valid.