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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 "contacts", "events", "tasks" or "mails" in BirdieSync Activation tab (not in ActiveSync/Windows Mobile Device Center).