According to your description, it seems that your father can’t stand the hot weather, which is pretty normal. Most people can’t tolerate hot weather for long periods of time. Hot weather increases body temperature. The thermo-regulatory mechanisms of the body atempt to keep the body’s temperature within the normal range of 36-37 oC, the most important mechanism for body cooling being sweating. When you sweat, water is excreted by the sweat glands in the skin and then evaporates. It is this water evaporation that, in fact, cools the body. Without evaporation sweating wouldn't be efficient. For example, sweating to cool the body is useless when the body is placed in a warm, closed space without ventilation and great moisture in the air.

In addition to evaporation, intake of water and electrolytes (salts) is also a very important when you sweat because the body’s water and salts are expelled through the pores. If water and salts are not supplemented, the body will dehydrate and body temperature will again increase, also known as hyperthermia. Dehydration combined with hyperthermia is called "heat stroke" and could be a life threatening condition if not diagnosed and treated on time.

Your father may report that he was hot and exhausted but was his body’s temperature really increased? Did he perform specific biochemical reports to identify increased hematocrit, decreased sodium etc... or did he only feel hot? It is one thing is to feel a sensation and another thing is to confirm those symptoms with a relevant physical and biochemical examination. The rest of your father's symptoms (“hard to breathe, his lips and arms go numb”) are not typical symptoms present during hyperthermia and dehydration.
Furthermore, depression can manifest with exhaustion and uncomfortable feelings.

Your father should request several physical and biochemical examinations such as measuring body temperature, blood pressure, electrolyte status, blood count, sedimentation, haemoglobin, hematocrit, blood glucose, hormonal status especially thyroid, etc.
You can consult an internal medicine specialist about his condition and request these diagnostics.