The house has ducts for a heater but does not have a condenser and coil.

I just had to replace an electric glasstop stove recently, and decided to spend less for the rental property and go with coil burner type than another glasstop.

Inadequate airflow through the evaporator coil can lead to this.

We bled the life out of a 25yr old compressor BUT I had already upgraded the furnace to a Hi Efficiency several year before keeping the old evap. coil in but I had a new evaporation coil ready to go. 

Duct work & a/c evaporation coils need cleaning, carpets & padding are usually toast as is most of the door trim the animal uses to exit/enter.

the vertical and diagonal cracks are concerning, OH is ground zero for foundation problems, I have stabilized several with horizontal cracking and bowing walls, but this makes me wonder if you have footing movement, I know of several homes here in WI that were built on unstable soil that has caused similar cracking, the houses often sink over time. there are methods to circumvent with piers or coils, my opinion is, if the deal is VERY good and you can monitor year to year and the cashflow is your play, its rare that houses fall down, consider the base value as static, dont count on appreciation in the future. you could be ok. 

HVAC includes the ductwork, and then the blower, and then the cooling system (Air cooling coils and outside condenser) or heating system (Furnace) or both that heats or cools the air the the blower pushes through the ducts.

FYI Frozen coil can be several things (link below).

I had to repair a roof leak+stucco, which cost me 5K this month, plus the dry wall with some mold on it from the leak, which is another $500, so 6K this month total.The tenant kept saying that her AC bill is super high ($650 per month during summer, its  3 story house and the ac is only 2 years old), so I promised her I will buy her one portable ac unit, which is about $300-400Then the AC leaked and I had an ac guy come in and he said the repair will be about $850 (so that is in addition to 5.5K this month) The Ac guy also said the coil needs to be cleaned and the ac filter is filthy.

A rooftop AC where the coils are leaking... an HVAC person is needed for that, not a roofer.