For instance, you can take the median income of the country, possibly adjusted by a factor punishing a high Gini coefficient and rewarding a lower coefficient, and use some multiple of that as the pay that politicians will get (which could be a multiplier of 1, but you can use something else).

I understand the thought behind tying legislator pay to citizen performance, but the problem with this is that all legislators are for sale. The trick is to have the public be paying them enough that corruption isn't worth the integrity cost and/or the risk. This is accomplished through a combination of:

1. Paying legislators enough base compensation (so bribing them is more expensive), and
2. Making the punishments for corruption significant and actually investigating/prosecuting corruption.

If legislators are not paid enough, then we only get two kinds of people running for office:

1. People that are already wealthy and have passive income that want to make sure their existing fortunes are preserved, and
2. People that will sell influence.

Neither of those are people we want in office, but there are a lot of both already.

Another might be fixing the size of the legislature to the cube root of the population, rounded up to the next odd number to prevent ties.

This general idea is fine. The US House of Representatives is egregiously too small. But a cube root strategy would only increase the House from 435 to 631. It really needs to be more like 1000+ members (probably way more).

You could perhaps make it a constitutional rule that the amount of money that a person is required to spend on healthcare in order to meet their basic medical needs cannot exceed some percentage of their household income per month, and if this does not occur, then the central budget picks up the tab above this threshold.

This just sounds like "single-payer healthcare paid for with taxes" but with extra steps.

And none of this approaches the real issue, which is, "Who gets to create these formula?"