You need to do a compression check to know for sure if you have a weak cylinder. As Ken said, on this model it very well could be the rack dampener screw that allows it to "romp."

A common cause for the climate control problem is the ignition switch. After starting the engine, wiggle the ignition key with the climate control set to full on and see if it comes alive. If so, the ignition switch is not too expensive and relatively quick and easy to replace. Don't buy a Climate Control Amplifier or any other part unless you have properly diagnosed and ensured that this is your problem.