Hi Ted,
if your site is using a SSO system you can still use the Moodle Mobile app, but the authentication will be done in the browser or in a popover window:
https://docs.moodle.org/35/en/Moodle_Mobile_guide_for_admins#Mobile_authentication
Once the user is authenticated in the browser or popover window, the app will be resumed and the user will be authenticated.
Regarding the second question, do you mean that you want to launch the app and authenticate automatically from another app/browser? In that case, the answer is no. The only way to authenticate users in the app is to launch Moodle Mobile first, then let it open the browser/popover and perform the authentication in there. If the user is already authenticated in the browser then he won't need to do anything, the browser will be closed automatically and he'll be authenticated in the app.
Kind regards,
Dani