You are correct in running this as a Ticket Task Report. The duplicate ticket name issue you are having is somewhat escaping me... Each task "resides" within a ticket, therefore if there are multiple tasks within that ticket the ticket name will appear in the report each time. In the report builder you are able to remove the "Ticket" column. The same amount of entires will exist but you will not see the ticket name at all.
Also I want to clarify some things for you, a ticket task does not have status. Rather it is measured but the percent complete. However the ticket itself does have a status. Two columns I would advise adding to your report are "Ticket Status" and "% Complete". You can then filter out around the completeness of these two columns.
Let me know if I can assist and further with this.