Can I use WordPress member login for app signup & login? Will my app users be logged in automatically after signing in once?

AppMySite is integrated with the core User module of WordPress. Users who register on your website will be able to sign in to your app as well, and vice versa. 

Users don’t need to log in to your app if they have already signed up or logged in once. Users only need to log in when they open it for the first time or in case they signed out earlier. You can manage additional onboarding settings on the User screens.

Read: Manage User Settings