When and where was the first "official" fan organized Lego Convention held?
According to the What is BrickCon? page, BrickCon claims to be the "longest running fan-based LEGO® convention". Its first edition took place in 2002.

Note that there are events which are older (LEGOworld in the Netherlands for example which I believe started in 2001), so I suppose BrickCon's claim is specific for conventions, and not exhibitions - although I doubt the line between both is very clear to start with (While LEGOworld is definitely directed to the public, there are fan-only happenings in the evenings)