The instructors were chosen by the Fleet Captain, normally you had a lot of time in the aircraft in position. The sim instructors were both OE LCA and SIM LCA....which I believe provided for more effective training, if something was not working in the sim, it showed up rapidly on the line and adjustments could be made to the sim training process. The max instructor pay was tied to the max monthly rate. If you taught in the sim I believe the max pay time was 80 hours, on the line as OE the max pay was 85 hours. Instructors had a pass perk of S3 passes. The pay was also based on what you held, not what you could hold.
To allow more than just a MSP culture in teaching the Instructor was given positive space to and from work and a hotel room.
Instructors were all Captains that were line pilots that taught approx 6 months of the year and flew the line the rest of the time...if there was a training crunch like now, we would have worked more. For the 330 fleet there was approx 40-60 instructors that did all of the sim and OE.

Like Nu said, only Line Capt instructors did full flight and any checking events.