Are there testers for thermostats?
It is service call experiences like these that give HVAC guys a bad reputation.

Many residential systems take between 8-13 pounds of refrigerant total.  It is extremely doubtful that you lost 3 lbs of charge in 3 days, so either the first technician charged the system incorrectly (left it undercharged), or the second technicians charged it incorrectly (left it overcharged), or they both charged it incorrectly.  If they actually put a combined 5 lbs of refrigerant in the system, then odds are pretty good that your system is now overcharged.  If so, that will hurt system performance and, depending on the type of metering device your system has, being overcharged could severely shorten the life of the compressor.  Secondly, the mere existence of a leak is not a valid reason for installing a whole new system.  If the leak occurred because the evaporator coil is corroded to death, okay maybe, but if there is a simple pin hole or leaking fitting then, no, that is not a valid reason for a new system.  Personally, I would not invite these guys back.  Sorry to be the bearer of bad news.

To answer your last question, I am not aware of any tester specifically for thermostats.  Usually some temperature observations and a few quick checks with a volt meter will tell a technician if the thermostat is the underlying cause of a problem or not, so no tester is really needed.  Also, with the introduction of advanced digital and wireless technology into the area of thermostats the variety of complex options has exploded.