A valuable feature of the Google Play Store is the ability to command a device you've signed in on to download an app without needing to go searching for the app on the device. I would like to do the same on iOS, but the iTunes Preview interface doesn't look like it supports signing in and cuing up app downloads on a device. Is it possible to tell an internet-connected iOS device to download an app from your computer either online or through the iTunes store browser?
Yes - iOS supports this out of the box with many MDM frameworks to push updates to iOS devices over the air. There are some free / open source MDM, but I would recommend something hosted like bushel that has better support and has been running a while.
If you need to host the MDM yourself, Apple's server app includes Profile Manager and would be my recommendation for your first server. (It's well documented, inexpensive, handles sertificates well and enables push notification registration.)