I will try to come back to this in future but some quick thoughts:>Should users be vetted (ie through Matrix) in order to ensure that the vote is not totally skewed by sockpuppets?
This should absolutely be avoided. It would be much more representative to have a system where mods read discussion threads (where people express a position and argue for it, and where mods can see IP hashes*.) and then take the arguments into account than it would be to have a system where only the people who care to join the Matrix can vote. (especially given the risks of the inverse: People who spend 95% of their time on Matrix voting on matters for the board, based on what they think would be good, without really posting on the board.
I'm not saying that should be the final system - I don't have a specific proposal yet, although the system used on Bunkerchan seems to have worked reasonably enough. (And I can say this despite having disagreed with some of the decisions made.)
In my view, the important thing is to be representative in practice - not in theory. Theoretically, everyone would vote in the Matrix and we'd have a neat honest system - but outside the world of theory it acts as a barrier to turnout, it serves to exclude voters who post exclusively on the board. Much better that their views are taken into account, even if there is no neat tally at the end of the day. If, indeed, a mistaken decision is made then there will be pressure to reverse it (Think back to the introduction of IDs for example) and the decision in question can usually be reversed.
*Yes, obviously people can use VPNs - but this is the advantage of the human element. It's one thing to have a thread full of IPs saying "no", "no", "no", it's quite another to have a thread full of posters arguing "yes, because 1., 2., 3., 4." "plus there is precedent in X, Y, Z on other sites…" "ah, but our circumstances here plainly differ…"
>Are anprims allowed? Post-colonialists? Anarcho-nihilists? Juche proponents, or Pol-potists? How about social democrats? And so on. To what extent are non-leftists allowed? Do they have the same 'rights' as leftist posters or are they treated harsher?
Cast the widest net possible while structuring rules around avoiding tedious culture war matters and avoiding the derailing of existing threads. Having an Anprim, SocDem or Pol-Potist thread where people who believe in those things can freely discuss them is one thing, having AnPrims, SocDems and Pol Potists come into an existing thread and derail it by the style in which they express their perspective is not.
Take the view, broadly, that for the site there are no enemies to the left of the status quo. This is good for PPH, good for discussion, and good for shifting people to the left. There should be no specific rules regarding the left, only general guidance for promoting good discussion.
As regards those to the right of the status quo, they will be more pinned down by rules around culture war nonsense. Allow them to make threads, but hem them in when it comes to acting outside those threads. Apply a less strict standard around derailment of right wing threads (while still, of course, applying best judgement - if someone's putting forward good arguments that we should nonetheless be able to counter, but people would rather shitpost, better to intervene to preserve honest discussion.)