1- Now this private blockchain is ready and if I have to add any new user does that mean I have to start a new server and on that server, I have to follow the above steps?
No. You are confusing platform concerns with application concerns.
Ethereum is designed so that all properly functioning nodes eventually reach consensus on the correct interpretation of transactions, and by extension, the system state.
The word "user" is overloaded. It could be an individual or an organization, a committee, a device. Ultimately, the meaning of it is an application concern that deals with matters such as authority and access control. This is expressed in contract design.
In practice, private networks often serve multiple organizations and those organizations may have an interest in ensuring others, e.g. competitors, don't have an unfair advantage. A topological solution is for each organization to operate a node.
Contracts are indifferent to which nodes users use to send transactions.
The resilience of the network, not speed, scales with the number of nodes. As more nodes are added, transactions/second does not increase, but the threshold number of complicit nodes required to revise historical facts does.
2- How do these wallets (ETH wallets) are setting up their infra? Do they have one server which is connected to public ethereum blockchain? and any new user who is joining/buying-ether from these wallets is basically getting added in these wallets as an account, not as a new peer?
Wallets and contracts do not have an infrastructure. The Ethereum network provides connectivity, processing, state, and resilience. When a contract is deployed on Ethereum, the code of the contract becomes an observable fact on the blockchain. Contracts must execute their instructions as coded. The public blockchain provides this platform with replication over many thousands of nodes. Published contracts require no further infrastructure support from their creators.
A private blockchain (usually) uses the same node software. The creator designs a topology that meets the requirements for replicas, secure access (each "user" needs to connect to at least one node), node-to-node connectivity, possibly behind a VPN, usually with at least one node in each participating organization so they can independently interpret the blockchain without reliance on competitors where incentives may not perfectly align.
3- From above two questions I have figured out that basically there are so many exchanges or wallets who are connected as peers to public blockchain and individual who is buying or selling using these wallets are basically accounts of these wallets. (added a new account eg: personal.newAccount("xux"))
Blockchain inverts the usual relationship of centralized data and dispersed nodes that request information. There is a shared set of facts and a protocol. That is all.
On a public chain, no further infrastructure is required. Private chain solutions reject public chains and use the (mostly) same software to create similar networks. It's analogous to the public internet, and internal intranets using the same software stack and protocols.
If you think of "coins" as objects or files that get passed around, it will never make sense. It's more like a spreadsheet in the sky that is apparent to everyone who can see it. There are strict rules about updates and an immutable history of all changes.
When a new node joins a network, the first thing it does is download the blocks, which contain a well-ordered transaction history and it runs the transactions itself, like a replay log. When fully caught up, the node will be in consensus with the rest of the network.
(Full) Nodes are similar to database replicas in that they contain the entire state history of all users and all contracts. The nodes agree on everything. It may help to think of contracts as similar to stored procedures. No arbitrary update is permissible. For example, a sysadmin is not allowed to INSERT INTO TABLE .... Everyone is restricted to send(data, {to: contract}) and the contract will decide if the update is permissible.
A rogue node would simply fall out of consensus with the rest of the network. The network would simply disregard the rogue node because the other nodes, having evaluated the transaction history for themselves would conclude that they disagree with the rogue and so should ignore it.
Consensus in a private chain has different constraints than consensus in a public chain. Consequently, variations of Ethereum such as Quorom exist. Design patterns for distributed are quite different from server-centric systems in either case, but transaction throughput and privacy matters can be treated differently than is the case on the public mainnet.
Hope it helps.
