Thanks for using the forum. I am happy to address your questions, and my answer will be based on the information you provided here. Please make sure you recognize that this forum is for educational purposes only, and it does not substitute for a formal office visit with your doctor.
It sounds like your husband was diagnosed with heat exhaustion and treated for it. Without the ability to obtain a history from him and examine him in person I can not comment on this diagnosis, however I will try to provide you with information about it. Of note though is that it is concerning that he is having headaches and left sided body weakness; it would be advisable to have some sort of imaging of his head if this has not already been done (CT or MRI).
Heatstroke results rom exercise or heavy work in hot environments combined with inadequate fluid intake. A patient with heatstroke loses the ability to deal with heat stress, such as sweating and temperature control, so they "overheat". Signs of heatstroke are very high temperature (more than 104 F), some confusion or personality changes, even a coma, and the skin may be hot and dry. Other signs include a fast beating heart, fast breathing, blood pressure abnormalities, lack of sweating despite a high temperature, changes in consciousness, dizziness/light-headedness, headache and nausea, and fainting.
As you can see, headache, nausea, and vomiting are signs of heatstroke. Again, while I can not comment on your husband's diagnosis, I will say it is concerning if he is persistently weak only on one side of his body, and if they have not already done CT scan and/or MRI of his brain, this would be important in order to rule out other potential causes such as a stroke, brain hemorrhage, or tumor.
Thank you for this opportunity to answer your questions, I hope you find the information I have provided useful, good luck.