The Primary Password is profile specific and is only used locally to encrypt the logins (never used with Sync apart from unlocking), so you need to set the PP on each desktop device (mobile doesn't use the PP, but uses OS means). I don't think that there would be problems with in what order you setup the PP, but I'm not sure whether Sync is aware of the PP is being used on other devices.

So once a PP is set on one device, how do other devices on the account react to that? Will a PP be automatically required on each subsequent device? "not sure whether Sync is aware of the PP is being used on other devices"

When you use the PP on a desktop device then you usually do this to have extra protection, i.e. if you have a laptop or other device that gets stolen or you simply leave it unattended then you do not want others be able to inspect those logins. If there are devices connected to this Mozilla Account that could use the PP but aren't using the PP then this can be a security risk. In the past this made it impossible to sync the logins to a mobile device, but currently mobile by default uses available OS authentication. So in principle it can be possible that you won't be able to sync logins from a device with a PP to a device that doesn't use the PP.

Cor-el, Thanks for your responses. I'm wondering how to get a more definite answer than "...in principle it can be possible that you won't be able to sync logins...". Once a PP is set on one computer, what exactly happens on the other computers? Each needs its own PP, right?

Yes, each desktop device can have its own Primary Password, it doesn't matter whether the PP is the name or not as the PP is only used locally on a device (the PP is used to encrypt the key stored in key4.db).

See Using a Primary Password with Sync.

https://support.mozilla.org/en-US/kb/using-primary-password-sync "Each device can have a different (or no) Primary Password, and they are not synced between devices." is the answer, thank you.