The process goes the same way in your private network as in the mainnet or Ropsten. You just have to define the endpoint properly to point to one of the nodes.
As it is a private network controlled by you, to deploy the contract for free you can just use a gas price of 0 for the transaction.
In ethereum there is no centralized certificate authority server like that of hyperledger fabric. Fabric is a private permissioned blockchain and all the participants need to be known. Hence the CA issues the identity.
Ethereum is a public network. Ethereum accounts can be created by anyone. Account creation in ethereum is basically generating a BIP39 ...
Since since the only POA consensus mechanism on Geth uses clique which uses GHOST , the only option is use a consensus mechanism with instant finality.
I would look at Quorum (IBFT) or Pantheon (IBFT 2.0)
In addition to the above, it depends on which flavour of PoA you are running.
Clique has determinstic finality and is works off GHOST (Greediest Observed Heavy Subtree). So depending on your trust model, you might want to wait a couple of confirmations.
Aura on the other hands provides finality after a majority vote i.e greater than n/2 validators have ...