Upon accessing User Manager, there are eight default columns: User name, Domain, Fully qualified name, Full name, Email, Comment, Language, and Locked. Is it possible to add/remove columns to display other key information we're tracking?
Not really unless you override the UserManager application. You can essentially reflect
Sitecore.Shell.Applications.Security.UserManager.UserManager and update the ASPX page located at \sitecore\shell\Applications\Security\UserManager (yes...it's not SPEAK) with your custom columns. Then, make sure that the application is using your new ASPX page.
The User Manager application is defined in the \Website\sitecore\shell\Applications\Security\UserManager\UserManager.aspx file.
You need to customize this aspx file to add new columns and the code behind of aspx file.
You should add custom GridColumn to the columns list of the Grid control. Sitecore used using third party component in the User Manager application. It is ComponentArt Grid control (https://www.componentsource.com).
Please refer to the ComponentArt documentation to find best way for filling in your custom column with the custom property values.