I have very little experience with card buildings (I've only built a few of Toposolario's shipping containers and Ikubes), so take this for what it's worth:
I've printed mine on 160 (g/m2) heavy paper, and they seem to be reasonably sturdy. I'm actually more worried about weight.. (Which can be remedied simply by glueing some washers to the bottom).
If you have a glue that doesn't warp the paper too much, you could perhaps double up on your paper?
Foamcore can be used as a backing too, just remember to either bevel the rear of a fold, or remove the layer of foam so you create a "slot" for the other wall to fall into.
Since you mention building on a budget, I'd advise to simply build one or two buildings out of 110# as is, with no extra reinforcements. Use these for a couple of games, don't bother being too gentle with them. That way you can determine just how much extra reinforcement you really need before investing in extra cardstock, foamboard and extra blades.