I will try to answer each item.
1. Yes, you need to make sure the shop has a capable tool and mechanics/technicians to properly utilize it.
2. Toyota’s service information refers to it as an air outlet actuator. The rest of us call it a mode actuator. A “blend door actuator” is also referred to as a temperature control actuator and it controls the door that adjusts the temperature of the air. The actual parts are sometimes the same part number but they are attached to different doors.
3. The actuator is considered an electrical part but many times it is a broken plastic gear inside the actuator that breaks. There can also be broken door shafts and linkage arms that would be solely mechanical repairs. This is where inspection and testing come into play.
4. Low voltage can cause many electrical parts to fail and should be avoided if possible. Typically a battery should be replaced about every three to four years regardless of how it tests. At least for a daily driver.
5. There are all types of scan tools with a wide array of functions. Obviously a true Toyota tool would be best. I would suspect that your shop has a fairly good tool since they could read the fault code in that system. There is a reasonably good chance that it can also read data and perform bi directional control functions.
There is a long shot that you could have a condition that we refer to as logic lock. What can happen when there is erratic voltage supply is that the computer’s logic modules can get scrambled. It is a fairly simple repair but there are some risks as well.
Since it is a Toyota product the doors need to be closed, the hood latch need to be closed with a screwdriver or other similar tool so that the vehicle believes the hood is closed. There may also be a push button switch on the edge of the hood that needs to be overridden. After the vehicle has sat in the above condition for at lest 15 minutes both battery cables need to be removed and the cable ends held together for a full minute. The battery cables can then be reconnected to the battery. Allow the vehicle to sit for a few minutes and then start the engine. Do not touch anything. Just let the engine run for about five minutes. Turn the ignition off for at least one minute and then restart and check the system operation. Hopefully this will correct the issue.
Now for the bad news. When this procedure is done it clears all learned memory in most if not all modules. This means that the idle speed may be higher or lower than normal. The transmission shift points will revert to factory settings and may seem harsher for several hundred miles of driving. The radio and clock settings will be lost. The radio anti theft code will need to be re entered if applicable. The auto power window, seat and mirrors may have to be relearned if applicable.