When should proof-of-work be done? Does it involve in mining a coin? Can it be done without real transactions to be recorded? I am trying to understand about what bitcoin is. I conceptualize bitcoin as a proof of work required when a new block (or new coin) is created. I also understand that a proof of work involves finding a value (nonce) that would meet a zero bits requirement when hashed. (please correct me if my understanding is wrong).
Am I right saying that bitcoin mining is to find such a value that would satisfy the zero bit requirements when itself only is hashed, or one that results in the required zero bits when the value itself and the transactions chosen, the previous hash, and other relevant items are hashed all together? If the latter is right, would finding a coin to create a block be only possible once there is a real transaction occurred? If this is true, what is it people mining bitcoins are doing? Are they actually creating and circulating a block in a peer-to-peer network with real transactions? If not, am I correct to think that they can be random people who use available computing power to find nonce values that meet the zero bit requirements and sell them to real owners of the nodes in a peer-to-peer network who use them to create blocks from their nodes. I am sorry for the dumb questions, but please help me to understand these fundamentals right. Thank you very much!
I also understand that a proof of work involves finding a value (nonce) that would meet a zero bits requirement when hashed.

The nonce is a value that a miner can change. It is one part of a block. The other parts are block headers and transaction details. The hash is of most of this, not just the nonce. The hash must be less than the current target &quot;difficulty&quot;.

Am I right saying that bitcoin mining is to find such a value that would satisfy the zero bit requirements when itself only is hashed, or one that results in the required zero bits when the value itself and the transactions chosen, the previous hash, and other relevant items are hashed all together?

The latter.
As I mentioned, the comparison is not the number of leading zero bits but a normal numeric comparison. The hash must be numerically less than the target.

If the latter is right, would finding a coin to create a block be only possible once there is a real transaction occurred?

A block can contain no real transactions other than the so-called &quot;coinbase&quot; transaction which pays the miner a mining reward.
Including other people's transactions allows the miner to collect and claim the transaction fees. SO there is a financial incentive for the miner to include transactions.

If this is true, what is it people mining bitcoins are doing? Are they actually creating and circulating a block in a peer-to-peer network with real transactions?

Yes.

If not, am I correct to think that they can be random people who use available computing power to find nonce values that meet the zero bit requirements and sell them to real owners of the nodes in a peer-to-peer network who use them to create blocks from their nodes.

No.