Do I understand this feature correctly as a pitch trim that is activated automatically whenever an extended pitch input (up or down) is commanded through the sidestick?
Not only for extended input. Any sidestick pitch input commands g´s, or a change in the flight path vector (it needs a normal acceleration to change the flight path vector). Releasing the stick does not mean you return to your previous trimmed pitch, but you maintain the new one. This is achieved by use of the trim.
To oversimplify a little bit, the Airbus Sidestick is the same as the trim switch on some conventional aircraft. Which is exactly what you need for a large transport aircraft. You need the stick to rotate, to level off at cruise altitude, to initiate the descend and to level of for touchdown. In the optimum situation, two of these actions are done via the A/P selector panel, so you basically need the elevator control twice per flight (both times in the nose up direction). So for some pilots that is all they have done with the stick in the recent years...
Back to AF447 and the THS being full nose up. If the pilot would have just inputed a full nose down on the sidestick, wouldn’t the THS trimmed nose down too?
Unless that would have resulted in an overspeed situation, and hence the system would not have allowed it, yes. So maybe it would have required to throttle down or use the speed brake until the system would have accepted this input.