The answer is and will always be same as you got on home repair forum "fan setting is for indoor fan,t-stat is ok,replace contactor on outside unit.

It is possible that your setting (low to mid 60's) is causing the coil to freeze, and shutting down on safety. This could explain why the unit needs to 'rest' for 40 minutes, in order for the ice to melt. Leaving your window open at night allows the night time humidity to infiltrate your apt. Leaving the a/c off all day allows the heat to build up. Both heat and humidity put a burden on your a/c which has to be satisfied before you feel comfortable. Being that your apt is 550 sq ft, the unit could be oversized which will result in frequent short cycling (less humidity removal). Try minimizing opening of your windows at night (unless outside humidity is low) and leaving your a/c on while away, although at a higher setting. You need to experiment. V

Anything is possible Crystal so yes t-stats often go bad same as other components. You are asking us to solve things that someone couldn't solve while laying hands on it. Adding to the near impossible task,you aren't giving much information for us to go on. Statments like "My ac never froze up" and "copper pipes outside are cold and wet like they should be"are misleading at best,outright inaccurate at worst. I don't have time to win debates and aurguments in order to answer questions for you. Get a different tech to look at it.

existing furnace is probably oversized...raising temp 4 degrees sounds good but a swing like that does nothing for comfort...I would change out thermostat when you install new furnace...you want an even temp setting to be maintained...add a filter cabinet at furnace return if you don't already have one...for a 1200 sq ft home besides the physical footprint needed, I would get a smaller furnace that can handle your 2 1/2 ton AC condenser...even a 60 KBTU furnace would be oversized. IMO