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.
X'raying would help establish if the heart is enlarged and if there are lung changes. As such, it helps stage the disease. Cardiac ultrasound would also be an option to evaluate the heart although it does not provide much information on the lungs.
Aspirin is often used after adulticide treatment to reduce the effects of the adult worms breaking up and circulating through the lungs. It can be used before adulticide treatment to reduce lung damage, but it is not routine. Perhaps your vet wants to delay treatment to allow resistant larva to mature and is using aspirin to minimize the risk of clots forming. I do not think giving aspirin if this is the rationale, does harm and it could help.
You might consider asking your vet why s/he wants to wait a month to start treatment and his/her reason for using aspirin.