Why does Google keep asking for user names & passwords I don't know?

Motorola phones use the Android system which is heavily reliant on Google software and network services. If you want to use these services (email-gmail, maps and so on) you will need a Google login and password to get access. If you don’t care about these services you can disable them individually in settings->Apps&Notification->Gmail(for instance). If you already use some other web mail facility (yahoo for example). You can find a yahoo mail app in the google play store (yes you have to have a google account for the play store also). Or you could just access yahoo mail via the browser that is already on your phone without resorting to a dedicated app. I have not tried this last suggestion and have no idea if the results are even sensible and usable on a small mobile device screen. There are compromises involved in using any “smart” phone/mobile device these days.