the download icons in the course can be enabled by clicking the top-right icon (3 dots) and then selecting "Enable download sections". Calculating which activities are downloaded and which aren't can be really slow, specially in big courses, so we decided to disable it by default because we had many complains about this.
The course download can be disabled in Moodle 3.5, but not in Moodle 3.4. It's done in Site administration > Mobile app > Mobile features > Disabled features, options "Download course" and "Download courses". If you have a custom app then you can just modify the code so they are not displayed.
That database error is returned by Moodle, not by the Mobile app. Please check that the database is running fine and it's correctly configured.