I have two computers, one that is significantly better than the other. However, the disc drive on the better computer won't read any discs anymore. I have the LA Noire discs, and if I downloaded it on the worse computer and tied it to steam, even if it was a nonsteam game, could I play it through steam on the better computer?
If i understand correctly you're asking if you add it as a "non steam game" in the steam libray will it follow to your other computer?
if that is what you mean then no. however you could use something like magicISO and copy the Images of the DVD's to iso files then transfer them to your other computer and use like Virtual Clone drive http://www.slysoft.com/en/virtual-clonedrive.html
to read the iso files and emulate a disk and you can play them just as if you had a CD in the drive