Yes, that is the upper reaction rod.
Just have a look to see where most of the movement occurs.
The ping/vibration damper only damps pedal movement but also reduces vibration reaching the pedal.
Knowing if it is faulty is a bit of guesswork, I think. Most people remove it as it makes the clutch pedal feel less positive. If you have a vibration, removing it won't help as it is meant to reduce vibration produced from the clutch/flywheel assembly. If it is only when the pedal is moving, the damper is ay fault. If it happens with stationary pedal, the damper may/may not be faulty.
I can't say for sure. Was clutch changed by you or a garage? If it was by garage, is it within 12 month warranty? That may be best. It could be the DMF. Too much throttle below seems to kill DMF. It is better to use a lower gear to try to keep nearer 2000rpm before using full throttle.