The core argument for x^2 is basically just that it's the only function with linear derivative, so for rational voters who calculate marginal benefit of additional votes purchased at cost x^2 (in terms of likelihood of changing the outcome of the "election"), you get a welfare maximizing outcome.

There's some assumptions underlying the theoretical result, but there's also a lot active experimentation in the real world, a few links: https://www.radicalxchange.org/concepts/quadratic-voting/, https://en.wikipedia.org/wiki/Quadratic_voting

Here's what I believe is the key paper on the topic https://www.aeaweb.org/articles?id=10.1257/pandp.20181002 (appendix: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2790624)

Lalley, Steven P., and E. Glen Weyl. 2018. "Quadratic Voting: How Mechanism Design Can Radicalize Democracy." AEA Papers and Proceedings, 108: 33-37.

It incentivizes the welfare-maximizing result. I don’t have a short “intuitive” explanation of why handy but the math is covered in section 8.3 of Public Choice III (should be easy to get in college libraries, pdf copies can be found online if needed).

There are a few mathematical reasons. I forget the details, but I remember that a key point is to consider marginal cost: the derivative of a quadratic is linear. I think the Central Limit Theorem is also relevant, as someone else here pointed out. Anyway, if you really want to know, you can read the papers!

Actually it's a good point and IMO for a population that may have a certain distribution of voting credits e.g. shareholders of a stock, adjusting the exponent can make sense to more fairly distribute power.