Payment processor question
Public key is point on ECDSA curve 

Public key + nonce != new public key


And if the above equation works, how can you calculate its private key !

Payment processors accept payment on their addresses, and send funds to users instantly, monthly, or send an equal amount of $ to their bank accounts.

I think you're looking for BIP32 wallets, in which addresses (public key hash) can be derived from other extended public keys (xPubs)