Doesnt helium kind of work like 2 and 3?. 1 person issues a challenge ,another challenges, others witness?. all split rewards accordingly?
No, because Proof of Something Else is not better than Proof of Work:
https://www.truthcoin.info/blog/pow-cheapest/1. Focus all mining work on the same block by getting all miners to work for the same pool, or create a super pool that consolidates the work of all mining pools. This would distribute block rewards most fairly but it would reduce energy consumption the least as the minimum work required for a share of the block rewards would only discourage the slowest miners from participating. This can be adjusted though.
Go on and do that in backward-compatible way. If you will do it in the right way, then maybe we will have some working example of that conception. But if you will do it in a wrong way, then it is 100% guaranteed that your project will be dumped. As I said in "mining decentralization-related discussions", all you need is collecting all shares and forming a commitment. With Taproot, it is easier than it ever was, so you can base your project on N-of-N multisig with all N miners, then try to make it simpler and cheaper.
2. Employ some kind of dice roll or rock-paper-scissor or pick-a-number scheme to select a miner at random who then does the actual work alone.
This sentence is some kind of definition of Proof of Work. We have pick-a-number scheme. That "number" is called "nonce", defined as "number used once". We cannot do "alone" part, because you cannot prevent other people from using computers, unless you want to shut down power grids and put citizens in ancient times with knights and horses.
the coins a miner earns would be proportional to the number of nodes he has in the selection pool
That is one of the reasons why the LUCK altcoin was destroyed:
https://bitcointalk.org/index.php?topic=5254068.4003. Yet another idea is to have nodes announce their intent to mine (register) and wait their turn in a queue.
This leads to centralization and depends on network connectivity. Also, it is not resistant to sybil attacks (or it may be, but then you have to turn on the Proof of Work).
However, the coins a miner earns would be proportional to the number of nodes he has in the queue, which would encourage node spam.
It can be attacked more easily, just by informing the network that you had a lot of connections, even if you were sitting alone on localhost. Newcomers will know nothing about the past without any Proof of Work. That is also another reason, why the LUCK altcoin was destroyed.
These schemes are only intended to replace competition as a selection mechanism. They do not replace POW.
Aha. That means some part is centralized and some part is not. Using Proof of Work here and not using Proof of Work there is not a good idea, because unprotected parts will be attacked, as they were in many altcoins.
So, to sum up: you can try to create some basic network for concept number one, with collecting shares, with Taproot, with N-of-N multisig, maybe on test network, maybe only with tweaked keys used as commitments, but forming a good concept for decentralized mining is hard. And there are some existing concepts like CoinPool, so maybe joining them is a better idea.