Hello Lele,
Could you tell me if you had duplicated events only on your mobile device and not in Thunderbird ? Were ALL your events duplicated or only some of them ? Did you have unresolved items ?
Are you under XP or Vista ?
If you are under XP, did you choose the "Combine" or "Replace" option during the establishement of the partnership ?
Had you appointments present on your mobile device before the first synchronization ?
First question: Is there a way to tell birdie to synchronize only one side?
Not for the moment. I hope it will be provided in a next version.
Second question: What happened to my events and is there an easy way to change that?
Did you choose the option "Link calendar to appointment and task category" ? If you didn't, at first synchronization if the "Combine" option was chosen or if you are on Vista, all the mobile device appointments will be stored in the "default" calendar.
Note: I suggest you to backup your data as mentioned in this 
thread.