Can I use Linux to download games that will be played on Windows?

Yep, totally possible.

1. Install Wine. If you primarily use Linux, you will likely have it already.
2. Install Steam for Windows on Wine, download the games you want to play. They don't even need to be playable through Wine.
3. Copy the games to Windows partition, as if you were copying them from another PC. You will probably have to tell Steam that there are Steam games in the place you have just copied to. Last time I have done this was around 3 years ago, but I know 100% that it's possible.

I did not happen to find any way to download Windows games from Steam for Linux.

Yes, but using a Windows virtual machine or Wine (the easier option and doesn't cost money, but compatibility isn't on par with native Windows installation) is likely necessary - distribution platforms such as Steam don't necessarily allow downloading non-native games (possible at least on GOG and Humble Store, though)