So if I purchase some songs from this store (e.g. someone gives me iTunes credit as a gift) I can still play them without iTunes? I don't need iTunes installed anywhere?
You need iTunes to buy (i.e. you can only access the Store using the iTunes app, not a web browser), but what you get is a completely standard AAC-encoded file without any DRM. I think pretty much any audio player available today can play that (as well as any mobile OS). Just drag the files in your library to your desktop or browse your iTunes folder to find them. After that you can uninstall iTunes and never need to see it again.
iTunes does embed your Apple ID (you need to sign up and get one to redeem your gift card, though you do not need to enter any credit card info or the like) and purchase date as well as time as metadata in the file, but that information is easily removed. There are no watermarks.
Is there some sort of iTunes kiosk, a publicly accessible device with iTunes installed where a customer could purchase songs, download them to removable media and then move the songs from the media to whereever they need to have them?
Or maybe you can use a friend's device with iTunes installed?
You can use a friend’s machine. Either create your own Apple ID there or redeem your gift card using your friend’s Apple ID and buy music from their account. After that you can just copy your music to an USB stick or to Dropbox or whatever.
Yes, you need iTunes to buy – but once you have the music, you do not need it anymore. They are just standard audio files that work with any (somewhat current) player.
