Most dogs that are clinically normal or have minimal signs of disease at the time of heartworm disease diagnosis recover completely and can return to their normal activity level. Your dog was coughing, which indicates that he has some level of heart and/or lung damage. Evaluation of heart and lungs with x'rays and/or other diagnostic tests might help predict response to treatment and long term effects. It is unlikely to give a definitive answer however.
Did the vet do any test to confirm the positive? False positives are not common, but do occur. Looking for microfilaria or sending a blood sample for confirmation is useful particularly if you are in an area where heartworm is not common.
Most vets treat for microfilaria (heartworm larva in the blood) either before or after treatment for adult worms. Ivermectin is commonly used.
Melarsomine (Immiticide Rx) is the usual treatment for adult worms. The drug has risks, but the benefits are much greater than the risk.
Aspirin reduces the ability of blood to coagulate by inhibiting platelet function. This reduces the risk of blood clots forming and blocking tiny blood vessels in the lungs and other organs.
Did the vet indicate why s/he preferred to wait a month before starting treatment? This is sometimes done in areas where winter kills mosquitoes. The delay allow larval forms of heartworm to develop to a stage where they are more susceptible to treatment.
Let me know if you have follow up questions.