Can I download apps via my Linux PC then install them on Android from there?
You can connect your device to the internet (using Wifi) and continue to use it just as you would with a plan.

So you can download as many apps as you want to your device using Wifi.

You don't need to be connected using the cellular network to access the Play Store (but do need the internet).

There are certain alternative App stores that are trustworthy, but be wary of downloading APKs directly off the internet.  There is no guarantee these are safe - generally, use the same caution you would use downloading software to your computer on your phone.  Only install from reputable sources.

You can actually pull an APK (Android App Package) off one device, and put it on another (using your Linux box), but this is not a task for a newb (you would need to install and use the developer tools)