Question: are there any other "schools" of AoPS's ilk that are worth looking into?

Nothing seriously comparable that I've managed to find, and I have been looking. Many less good things. I've been watching what's provided by the DaVinci group (organisation has been hopping around with funding, current page

here) but haven't joined/used it yet. I see they have added some maths provision, OxMaths, since I last looked, but it's not suitable for your kid or mine.

Maybe he could do some research.

Ah yes, the "grow your own collaborator" plan. DS wants to prove Goldbach's conjecture; we'd far rather he proved P ne NP (and not only for financial reasons), but we'd settle for Goldbach if that's what lights his fire ;-) ;-)

One concern is I was wondering if taking university courses (while officially being a seconday school student) could disqualify you from competing in certain maths competitions.

This is a valid concern, I think, and the rules are quite likely to change given the fluidity of the current situation, so it's one to watch. For the IMO at present,

Contestants must not have formally enrolled at a university or any other equivalent post-secondary

institution, and they must have been born less than twenty years before the day of the second Contest

paper.

Unfortunately, "formally enrolled" is not further defined, though some countries (Canada turned up on my google) elucidate this as meaning enrolled on a degree-granting programme.

So after finishing calculus in 6th grade, our son won't be twiddling his thumbs for the following 6 years waiting for the next maths course to show up at uni. There's plenty of maths he can do in the meantime, even if it takes some scrambling and improvising. If he has to repeat some material, hopefully it's at an elite (Ivy League/Oxbridge type) university, where it's presented at a much higher level. (I was looking at the Princeton University website once where it said words to the effect, my paraphrasing, "yeah, sure, you mighta taken calculus before, but you haven't taken our calculus", and they do have a point.)

They certainly do (and if you were talking about Oxbridge literally, the course assumes you've done plenty of calculus anyway, since it's on the normal school syllabus here rather than being nominally university maths). All the same, if much of his six years after Calculus BC turns out to be spent doing university-level analysis courses and research in that field, he could still easily end up more suited to teaching Princeton's intro calculus course than taking it... but here we surely come to "plans are useless, planning is vital".

By the way, does anyone know what it takes to get into maths at an elite institution? Is it based purely on merit? Or do you, as some have suggested on this forum, have to fluff your CV with extracurricular activities like volunteering at the homeless cat shelter and playing polo?

It will surely depend on which elite institution, but I can say for sure that neither Oxford nor Cambridge could care less about anything but academic merit, because they're both on record saying this clearly. I sort of doubt that someone who had IMO medals and/or papers in reputable journals to their name, and didn't have two heads, would in practice get turned down even at US elite institutions - but it would be good to hear from someone who knows.