What is the key generation process?
As Nat says in his answer, the keys are secp256k1 or secp256r1 curve keys. You can read more about that here: http://www.secg.org/sec2-v2.pdf
Can the same keypairs generated for the mainnet be used on any private network?
Yes. A private/public keypair created for one EOSIO chain is usable on another chain.
What is the ...