How do I troubleshoot the issue of Outlook not properly saving or syncing my flagged emails?

Check your Outlook settings and verify that your flag settings are enabled and/or configured correctly
Reset your Outlook cache
Create a copy of your Outlook .pst file
Create a System Restore Point
Test for any interference from any enabled add-ins or plugins
Run a virus/malware scan
Verify whether you are running the most up to date version of Outlook
Check your storage location for Outlook emails
Check that Outlook is compatible with your Operating System and other installed programs
Change your Outlook folder structure to view flagged emails in their own folder
Perform a selective synchronization in Outlook