Here are some prior responses to that sort of question FYI - generally, if you have high humidity in your house or furnace/air handler area, you can get mold. Also, A/C evaporator coils will mold up, especially if they get pretty dirty due to poor filtration, also if the A/C is short-cycling (running only a few minutes each cycle) which makes the condensor coil get wet but doesnot run the blower long enough to evaporate it off. Too cold a coil can also cause this (leaking or low on refrigerant or running at too high a pressure drop) so it ices up, then stays wet after the cycle is done and eventually turns to wetness that promotes mold growth. A coil drain tube/pan that is not draining properly can also quickly promote mold growth, so your HVAC tech should check for all these things.
The coil is usually the first thing to inspect for mold - can be cleaned in almost all cases, so a Heating and A/C contractor (A/C specialist for type of system you have) would be your normal first step. If you have severe mold in the ducts, or a lot of moldy dust buildup, then a duct cleaning job may also be recommended. In severe humidity areas (not your case) sometimes (like in parts of the deep south and east coast) chronic moisture in the ducts can lead to perennial mold issues which require ultraviolet disinfectant lighting at the coil areas (commonly front and back to treat both areas) to control them permanently.
In very dusty areas (hello, lower elevations of Arizona like yours) commonly you need a pre-filter to remove dust as it comes into the air handler area, then a HEPA (high efficiency particulate air) filter refiltering that incoming air plus filtering all recirculated air in the system before it goes through the air handler - that removes the growth media and food for the mold.
In your case, since you have a heat pump, evidently he said the heat pump air handler (fan and coil and the box they are in) is crudded up and needs a thorough cleaning - which, with some models which are poorly designed and not planned for cleaning, does require removing the motor to clean it - or sometimes the motor is so clogged up it needs to be removed to clean it, which might be your case.
Since he blamed the dog on it (at leat partially), sounds like your filters are not in the correct place, you have return or makeup air bypassing the filters, or they are not very efficient - because dog hair is one of the easiest things to trap with a filter. Or perhaps you have not been instructed on how to check the filter and change it when needed - typically every year and least and commmonly every 3-6 months in houses with animals that shed a lot, so it plugged up and started pulling dirty air around the edges of the filter or through openings in the ductwork.
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