In a private Ethereum setup:

  • How does a node that computed the nonce understand that 51% of the nodes have confirmed the mining?
  • Is there some kind of a counter maintained which shows number of confirmations?
  • Are the nodes which confirmed the mining rewarded with some amount of Ethers?
  • Does a node confirmation mean that along with confirming the nonce, all the transactions in the block are also confirmed?

It seems you have mixed up some key concepts in Ethereum. Let me try to clarify these.

Nodes do not need to care about how many confirmations a block has. A confirmation is a term which is useful only to someone who accepts Ether as payments: if a transaction has 5 confirmations it typically means there are 5 blocks mined on top of the block which includes the transaction. 5 confirmations is quite much already, but it's up to the Ether receiver to decide what number of confirmations is enough to be "sure" that the transaction won't be reverted.

There actually is no such thing as "51% confirmations". The percentage 51% is only used when talked about different attack vectors against the EVM: you would need 51% of the network's hashing (computing) power to be able to attack the network with double-spend attacks.

Furthermore, mining is confirming. When a block is successfully mined it means the block is confirmed and added to the (thought-to-be) canonical chain. But yes, mining (confirming) a block means all the transactions within it are confirmed.

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