Do the rear cog, chain, and chainring (chainwheel) all have to be the same size? 1/8" or 3/32" the whole way? I'm running a 1/8" SRAM chain on a 3/32" Shimano rear cog right now.

No, but you'll have some slop. I was running a 1/8" chain on 3/32" rear cog and chainring for a while, but when I mashed down on the cranks, I could feel the chain slop from its left limit to its right limit.

You can go from bigger to smaller but not vice versa. Same principle as running a long-cage r.d. on a double but not being able to run a short-cage r.d. on a triple.

You're probably feeling something else. That is 1/32" of an inch (less than 1mm) of lateral play, that you are claiming to feel linearly.

I've run 1/8" chain on 3/32 cogs for a long ass time. It's a tiny bit noisier (in the way that a small fly is noisier than a mosquito) but other than that, it works just as well.