I selected contact, calendar, tasks, or e-mail synchronization in ActiveSync but nothing is synchronized.

"contact", "calendar", "task", "e-mail" are Outlook categories. If you selected one of them in ActiveSync/Windows Mobile Device Center options, you will synchronize with Outlook, not Thunderbird and Sunbird. If you want to synchronize with BirdieSync, you must then select "cards", "events", "tasks" or "mails".