Let's say we have a network with n nodes in it. Is it possible, when the hashing power of the network is calculated after a proof-of-work, to accurately estimate n with an algorithm taking the calculated hashing power into account?
There is no correlation between hash-power and number of validating full-nodes in the network.
Consider the following:
Hashrate is necessarily anonymous. Therefore, there is no way to determine how many miners are operating on the network. Furthermore, a miner may operate an arbitrary number of full-validation nodes, from which it creates its candidate blocks.
It is possible to approximate the global hash-rate from recent block intervals/difficulty, but this only captures hash rate that was working on the current strong-chain, not hash rate working on another branch (e.g. selfish mining). Even if we could approximate global hash-rate, it has no relation to the number of full-validating nodes.
Hash-rate is proportional to the demand for Bitcoin transactions. Higher confirmation demand will lead to higher fees in the fee market, resulting in new hash rate joining the market. One may assume that that higher confirmation demand may lead to higher full-validating node count, but that too is not guaranteed, if the node-validation is too costly (big blocks) and becomes centralised (custodial services).
I don't think so. This would only hold true if every node would be mining. But in the current situation of the bitcoin network most nodes are just verifying the blockchain and relying blocks and transactions. Only a few nodes are actually mining. They run a "mining server" which distributes mining problems to the mining hardware in their mining pools. So no you can't really map hashing power to network size.