Since Plex has now added streaming content for non-paying people the parental control are now virtually useless. Here is my scenario. My kids have Kindle Fire tablets so that I can get them e-books and they can listen to music from our plex server. That’s it and that was great until the new streaming stuff was added for non-logged in users. Now if my kids happen to accidentally (or purposefully) uninstall the Plex app and they re-launch it they can stream whatever they want. I happen to be a good parent and I keep tabs on these things, but it would be very easy for this to get abused. Not to mention that yesterday this happened and I had to log back into my Plex profile on the kids tablet and what am I presented with FIRST THING? A mostly naked picture of a woman from the cover of the movie Cashback. That is SO not kid appropriate. Why have parental controls if they are no longer useful? I would suggest a few things. Not allow managed users to log out, persistently store the login so that if the app gets uninstalled when it is re-installed they will be reconnected to the subscriber account. Also, potentially determine if the app is being run in Amazon FreeTime mode and not allow streaming content and certainly don’t show them picture of mostly naked women.
It doesn’t address your other points, but if the only reason they are allowed to use Plex is music, you might look at Plexamp.
Just stumbled across this post.
Putting aside it’s not so much a feature suggestion…
If you want your kids to access those online media sources offered by Plex, you should look into the documentation on parental controls and how to apply (age) restriction profiles. Alternatively you might want to consider disabling those online media sources for good.
If your kid has a managed user account, they cannot simply sign-out from an app.  So media available while the app is not signed in should be of no concern.
If you don’t want your kid to uninstall / re-install apps, you need to reflect this in the parental controls of your Android platform (e.g. Amazon FreeTime).
That’s not how Amazon FreeTime works. My kids don’t have the ability to install just any app. They only have the ability to install the apps I have added to their account. There is no way to prevent uninstall with Amazon FreeTime.