Apps purchased from the official Google Play store can only officially be installed through having the phone itself connect to the internet. You can buy the app from the web site and tell it to queue an installation to a device, but officially the device itself must be connected to install or update apps.

Android does support unofficial app distribution as well though. If someone gives you a .apk of their app you can just copy it to the phone and select it from the file browser to install it. You can update apps installed from the Play Store in this way as well, just the first install needs to be done the official way to get the license loaded.

Music is just files, you can copy them on and off the device in basically the same way as an old MP3 player unless you bought from a DRM-based provider in which case they'll be "protected" somewhere else.

There's no real need for an iTunes-like setup with Android. Your PC has no need to be involved in the app installation process and everything else is just normal files you can manage with the file manager you already have.

I pay for Google All Music Access and it lets me get any song they sell normally and stream it as well as download it for offline listening. Has worked wonders for me. The only thing I wish is that my Android phone had the 3 audio buttons I miss from my iPod Classic: play, reset / go back a track, skip track. At least my steering wheel lets me do it...

This doesn't tick all of those boxes but in case you don't realise - in the last few versions of Android I've had installed I've been able to set a long press of the volume up/down hardware buttons to act as next/previous track. I can currently do it through settings/buttons but not sure how it worked in the past, it might have been app-specific

Yep, can buy apps in a browser on your computer, and have them install on your phone. Music is a tad different, Play music on the phone can Auto download music it thinks you might like, and I would guess that it might do that if you bought a particular album.

Or if you want more manual control, Play Music can also be set to download the contents of a playlist (or artist or etc.) Add your new album to that playlist and it'll be downloaded.

Isn't that possible with play music with every Android version since (I think) 4?

The latest iTunes update removes the possibility to handle apps though.

I just use Groove on both android and windows. All my music is synced to both places and I can get music I acquired elsewhere onto everything by just sticking it in OneDrive

and you can get groove subscription for free by participating in the microsoft rewards program!

though its 70k points/year makes it the most expensive reward.

There is a glaring downside to that service though. I can't open the website to stream the music from ubuntu. The website just doesn't load.

Not sure, but maybe play music can do this?