Hi @requiredusername. You should receive push notifications when your Ring Alarm system is alarming. In the Ring app, tap the menu in the top left > Settings > Alert Settings > Alarm and switch the toggle at the top to Push to adjust your push notifications. From here, you can control all of your alert options for the Ring Alarm system. If you are not receiving push notifications despite your settings, please review the troubleshooting steps in this Community post.
As for the Schlage lock, any automatic locking features would be on the Schlage side specifically as this is not a feature on the Ring app. I’d suggest reviewing what settings and features are available in the Schlage app for the specific model of Schlage smart lock you have.