Your question is very broad and can't be easily answered. There are few existing products, so the answer is also somewhat speculative.
Aerogels are a form of material, like foam or dust. It's very likely that a suitable house insulation could be made with an aerogel, almost regardless of the detailed requirements that you have in mind (barring contradictory requirements, like "cheap" and "aerogel"), since a wide range of aerogels already exist and some could be used this way.
Current aerogels are typically made from silica gel or carbon particles partly fused together, but could in theory be made from almost anything. You could conceivably replace the aluminium in an alumina aerogel with another metal to get different characteristics, since those are essentially just fused dust (in the same way as humans are essentially just bags of dirty water). The difference between a sintered metal and a metallic aerogel is largely density if we ignore the difficulty of producing it. Silica aerogels turn to dust when crushed, as well.
Since aerogels typically have a much lower thermal conductivity and density than their parent materials, they can be used as insulators. Flammability will typically be higher due to the larger surface area and likely oxygen infiltration over time, even if oxygen was not present when the aerogel was made. Toxicity depends almost entirely on the constituent materials, with silica aerogels being largely inert and minimally contaminated. But carbon aerogels typically burn, and if you made an aerogel with fused plutonium dust it would be dangerous in all the ways that plutonium is.
For home insulation with current technologies, silica aerogels are the best candidate. They are expensive, and somewhat fragile. Unfortunately the current methods for producing them are inherently expensive (supercritical drying at high pressure and temperature).
This product from the site above sells for $2.30 a square foot when 5mm thick, with an R value of about R2.8 W/K/m² (if I've done the maths correctly from 14 mW m-1 K-1), compared to a similar thickness R1.3 foil+bubblewrap for $0.40/square foot here in Australia. The aerogel is also fragile and dusty, where the foil product is not. A layer of foil plus a layer of "earthwool" R2 for a total of $0.70/sq foot and ~50mm thick gives better insulation for a significantly lower price.
I think it's more likely that an affordable aerogel will be made from a carbon lattice, since we have a multitude of different techniques for working with carbon already. Current carbon aerogel thermal insulation (see link below) is made via pyrolysis (combustion) so is usually fragile and chemically reactive, limiting it to specialised applications (most vacuum-related).
The main problem with using aerogels for home insulation is the variety of low-cost solutions already in the market. At the extreme are things like straw-bale insulation, where the raw material is a low-cost byproduct. Many cellulose-based insulators use similar source products, making them equally cheap.
Which is also the problem - to be cost-competitive with existing insulation products the aerogel would have to be better in some way and not significantly more expensive. Many parts of homes don't have significant thickness limitations - it's quite reasonable to have 200mm or 300mm of insulation in walls, and if an aerogel competitor was available that allowed that R value to be achieved in 50mm that would not be a huge advantage.
Here's a pdf presentation (from 1999) on carbon aerogel thermal insulation.