Is signature/private-key required to accept payment over Lightning Network?
Yes invoices are signed according to bolt 11. One reason we do this is to fight dos attacks. If I would not have to sign my invoice I could create an Invoice for any node. That node would not have the preimage for that payment hash and routing of the payment would fail all the time. 

While I can imagine other processes that don't require a signature in the invoice we still need the private key / signature to update the payment channel on which the payment is coming in. This is needed to accept the offered htlc and to settle it. Since we certainly need a signature for the htlcs we can use them for invoices as well.