CiviCase - remember user choices in search filters for activities
The setting Preserve activity filters as a user preference present in Display Preferences is limited to only Activity tab displayed on the contact summary page. As of now, there is no way to save the user choices in case activity tab.

Similarly, to select Scheduled as a default option for status, you might need to write a minor customization in an extension to make it work. Either using the buildForm hook and setting default value OR using a smal js code snippet.