what kind of sheet-metal screw do I need for wood to 1/8" steel?
I'd consider 1/8" in the "thicker" category.

Thinner would be like steel ductwork, thick enough to receive &amp; hold screws, but thin enough to use an ordinary sheet metal which just has a pointy threaded tip (looks similar to a wood screw).

I have recently drilled wood into steel on a few different projects which was about 1/8" thick and used screws like this:


Not pushing any particular brand but this type did work well for my application. The key element of the screw is the tip which is similar to a drill bit. The flutes of that drilling tip have to be at least as long as the metal is deep.

A tip is to make sure your screwdriver bit fits the screw head really well. Since it takes some time for the screw to bite (compared to screwing into wood) you need a good solid connection to your driver. If the wood you're attaching is thick enough that's not likely to be a major problem, however.