Does this Chromebook support Google Play apps?
I don't have the "shopping bag" icon and if I got to play.google.com and try to download something, I get an icon indicating that I don't have any devices.
No. A Chromebook runs its own version of apps. The Chromebook's operating system is a proprietary version of Linux, that is so locked down that it doesn't even allow you to remote into it (which is what I want/need to do to help my mom when she gets stuck).