1: Now that we've narrowed down to Sun and Sand, what areas and roles do these two weathers struggle or lack in? Are any of these holes in their team frameworks shared?
As has been demonstrated, Sun needs speed control and Sand needs breakers. Particularly
special breakers.
2: What typings address the weaknesses of Sun/Sand teams individually?
Firstly, Sun's best abilities are stuck on grass types, which are weak to the fire empowering that Sun provides. As such, I think that we need something that is not weak to fire (resisting it would be even better). This suggests Rock, Dragon or Water (water is not ideal, due to preferring Rain to Sun though).
Secondly, Sand's main two mon are both weak to Fighting. A Fighting resistance, or at least, neutrality could help as could a type that does SE damage to Fighting. This suggests Ghost, Psychic or Fairy.
Scaring away Pelliper and Alolan-Ninetails would help (with that order of priority), which suggests Rock, Electric (though I don't want CAP24 to benefit from Rain's boost to Thunder accuracy) or Steel.
People are also understandable scared of Ash-Greninja (Suggesting electric, dragon, grass, fighting or bug and discouraging types that are too weak to water), Tomohawk (Suggesting electric, flying, Psychic, poison or ghost and discouraging typings weak to Fighting and Flying) and Landorus (suggesting Ice or Flying and discouraging types weak to Ground, in particular, and Flying, to a lessor extent).
3: Is it better to have a different typing than the common Pokemon on current Sun/Sand teams, or to stack typings with existing abusers?
The only benefits Grass supplies are STAB Solar Beam/Blade (which cannot be relied on, due to weather switching), water resistance (which is supplied by Sun anyway) and Ground resistance. It comes with a bunch of problems though, like weakness to Ice, Flying and Fire. I think that most Grass typings should be avoided unless paired with something that helps deal with those.
Fire helps with Sun's boost, meaning a lot of power, but Sun needs speed more than power, so this might be less necessary.
Sand needs Rock, Steel or Ground to avoid chip damage, but abilities can overcome this dependency and eating chip damage can soetimes e worth it.
Those typings also gain a Sand Force boost that an additional STAB boost can lift to the large power needed for a breaker.
However, Sand Force also mitigates chip damage from Sand, and using it for coverage instead of STAB makes our breaker more versatile, which could be worth it, even at the cost of power.
EDIT:
Professor Mesquite We don't need to worry about other Sand or Sun teams. If they counter us, they've become relevant and we've done our job.