The "tick-tick" that you're hearing is coming from the DVV (Dual Vacuum Valve). See if this makes any sense and I'll try not to write a short book in the process:
The DVV controls the position of the power servo stem by applying and bleeding off vacuum from the power servo diaphragm. The setting of the temp. control lever establishes the baseline position of the blend door on the interior air box outlet which is mechanically connected to the power servo stem. The DVV is switched On, applying vacuum to the power servo diaphragm, and then switched Off once the correct position of the blend door is established. The problem occurs when a vacuum leak allows vacuum to be bled off of the power servo diaphragm. The vacuum leak allows the power servo stem to move slowly to the left after the DVV is switched Off, which in turn causes the A/C amp to turn on the DVV once again in an attempt to re-establish the proper position of the power servo stem, which it does, but only temporarily due to the vacuum leak. :roll:
The DVV contains two very small vacuum valves that are operated by two very small solenoids, all internal to the DVV. One valve is used to control the supply of vacuum to the power servo diaphragm while the other controls the amount of vacuum bleed off from the power servo diaphragm. Am I making any sense yet? :lol: Check for any vacuum leaks in the DVV/power servo section of the auto A/C system. You may have a leaky vacuum hose, a leaky power servo diaphragm, or a leaky bleed valve in the DVV (which is what I've found in the past). Try swapping out the DVV and see if that cures the cycling problem. I've seen one case where the DVV bleed valve was to blame. The valves are a needle and seat type arrangement that resemble a scaled down version of the needle and seat in the float bowl of a carburetor (only the needle tip is hard plastic instead of butyl rubber or Neoprene in the DVV). :? Repairing the crummy valve needles and seats in the DVV has proven to be a less than enjoyable experience for me, and I highly recommend replacement of the DVV with a new one instead! IOW, don't attempt to disassemble the DVV unless you have steady hands, nerves of steel, and much patience to deal with the several tiny springs and steel check balls that lurk inside. Chuck the DVV and buy a new one!
The reason the "tick-tick" stopped cycling and ended with a single tick is because your engine wasn't running and thus no continual vacuum supply to the DVV. The more erratic ticking sound that you're hearing when you move the temp. lever to the far right is due to poor contact in the temp. lever potentiometer. Giving the potentiometer a shot of contact cleaner should improve or even cure that problem. The contacts tarnish over time and make poor contact which drives the A/C amp and the DVV crazy.
HTH!!