I have a how to question regarding an old Nexus 7

I'm sorry to inform you, that a factory reset will not downgrade your OS.

 

Youvwill have to go to your phone settings and set it to download from unknow source's. Then choose an alternative App/apk market.


https://m.apkpure.com

and/or

https://www.apkmirror.com/apk/vojtech-horanek/apkmirror-unofficial/apkmirror-3-5-release/

 

yes just do a factory reset and your account will be erased. it will not downgrade the device...it will just remove the data off of the phone.

 

To remove your Google account from this phone you need to go into the Settings >> Accounts menu:
https://support.google.com/android/answer/7664951?hl=en

As mentioned in previous posts, a Factory Reset will not change your phone back to an earlier OS version, All it does is wipe the user data, it does nothing to the operating system. The internal storage of your Nexus 7 is divided into several partitions with most of them dedicated to the installed Android OS, and largest one is set up to be the user data partition. A Factory Reset only wipes that user data partition clean, it doesn't have any affect on any of the system partitions.

You referred to the phone being unstable after a system upgrade. Did you try wiping the system cache partition? That's a commonly recommended thing to try after something like a system upgrade (note that's becoming more of legacy thing as newer versions of Android devices no longer have separate system cache partitions).
Anyway, wiping the system cache isn't something done in the Settings menu, you need to reboot your Nexus into its Recovery Mode, instructions on the process here:
https://support.google.com/android/answer/7664951?hl=en
Remove your Google account in the Settings menu, then reboot it into Recovery Mode to do the wipe system cache partition and a Factory Reset. Then after restarting normally see if the phone is running normally again.
At some point you'll probably want to set up somebody's Google account in order to be able to access the Play Store.