To answer the questions - Staff reads the forums and if they see a thread that needs moved, or if a thread that's reported as needing moved actually does need moved, they'll move it.
Graveyard is for spam, non-discussions, and threads which the OP requests to be graved (typically when they realize and, more importantly, accept that their thread was a bad idea)
As far as who does the moving, it's whatever member of staff deigns to. Maybe one staff member thinks a thread is fine as is, and another decides it's more an achievement than a General Discussion post and moves it, for example.
I imagine scrutiny is applied more liberally to those who tend to post against the rules more often. Maybe also for those who a member of staff holds a grudge against. Staff are only human, after all.
A new or uninformed player would know someone is staff by seeing their name in the staff list or by recognizing their role as staff.