The stats only list when someone who had anything to do with OAA (i.e. did the ground-school only or the full Integrated) changes job.
It's not restricted to your first job.
Every time you change job you will reappear on the stats page - a bit misleading really.
I'm not sure that's correct. As far as I'm aware the statistics are the same as the ones published in their building which are listed course by course (integrated and modular). They can't/don't publish your name and course twice if you have changed jobs...
In terms of KLM and Qatar - they start off as SO's. Cathay, as stated above, you go through full training again. Not sure about Singapore...probably as a SO though.