Pilot DAR, interesting write-up.
I have owned two stabilator aircraft; a PA28-140 and my current Beech C23 (180 hp Sundowner). I have had a couple of surprises on landing the PA28, with full aft stick, and it appears the tail stalled and the nose wheel came down rather suddenly and the landing was more or less a "three pointer" (NOT good on a tricycle!).
I've never had it occur on the Sundowner though. And I have also never experienced the touchy behaviour the Sundowner is supposed to be reputed to have on landing. However I have one of the rare aerobatic models and there is more authority to the stabilator on account if there being a leading edge fillet between the fuselage and the stabilator, which isn't there on the non-aerobatic ones. As it happens the Sundowner is the easiest aircraft I've ever flown for landings (nice quick ailerons in flight as well), as long as you stick very close to the numbers (80 knots with no flaps or 10 deg, 70-75 knots with 25 deg flaps, 68-70 with full flaps), and fly a power-on approach. But I digress!
Still, I agree with you, stalling a stabilator is possible. Didn't in fact Cessna have to modify the stabilator on the 177 with slotted leading edges to avoid this possibility?
For the PA28, which is the subject of the original question, the trim tab angle relative to the stabilator changes with stablilator position relative to the aircraft.
Minor quibble, but it isn't a trim tab, it's an anti-servo tab. Without this tab, a stabilator provides no control feel, contrary to a traditional elevator where drag increases as the angle of deflection increases; as you note on the stabilator the whole thing moves and the airfoil section remains constant. The anti-servo tab is meant to add drag and increase control forces in proportion to the position of the stabilator, to give it artificial feel. Of course, it also provides the trimming function but that's not its primary purpose.
My "I learned about flying" story: one year I had my Sundowner fresh out of annual and went flying to do the Transport Canada mandated post-annual test flight. The anti-servo tab attachment bolt to the actuator was not re-installed by the mechanic and I didn't catch it on my pre-flight. I took off and sure enough there is NO control feedback without the tab working. This is actually very dangerous and can lead to inadvertent over-stressing. I figured something was whacky and beat a hasty retreat back for a landing when I discovered my (and the mechanic's) oversight.
Beech