Is it possible to add "total time spent" to the stories backlog as a column, on the story level and on individual tasks and bugs? What about on the Kanban board?
Yes, this is possible.
TFS Timetracker can automatically fill a field from Team Foundation Server or VSTS with the time spent on this particular item. This is possible with the "Completed Work" feature.
First, you need to have a field in TFS, either an existing field such as "Completed Work" or you update the work item template with your own custom field.
Then go to the "Configuration" tab of Timetracker -> "Settings" -> "Work Item Settings" -> "Completed Work" -> click the "Enable filling in total tracked time for a work item field" checkbox -> select the field that you want to use in the dropdown and then click "Apply".
Timetracker will then fill in this field automatically and you can use it everywhere in Team Foundation Server or VSTS.