3/32 is standard mountainbike chain? 1/8th is the the burlier BMX style?
Yup, that's correct.
You can run 1/8" chain on a 3/32" front chainring no problem. There's a little bit of lateral play but works fine. I'd recommend an unramped DH or SS specifc chain ring, although truth be told I've run with regular ramped chainring with no issues.
However, DON'T just take apart a cassette and use one of the ramped cogs on the back. That shit will skip the first time your chain is slightly out of adjustment and you crank on the pedals hard, and you'll smash your knee/balls on your stem. Bad times.
You need a proper SS cog on the rear. Cassette cogs can't properly hold chain without a rear derailleur tensioning it (even when they're new, but especially if you're raiding an old cassette. That 16t cog is probably the most worn out of the whole cluster).