A. If your cat is vomiting there could be several underlying causes. I guess the first thing I would want to check is the thyroid level, since I have definitely seen cats that were at one point "controlled" on a specific dose of medication no longer be controlled, and the dosage has to be adjusted. This is why we always recommend rechecking thyroid levels yearly, even in hyperthyroid cats that are clinically doing well.
If the thyroid levels have recently been checked and are stable, then I'd start looking for other causes, such as GI disease. Other possibilities include kidney disease, which can definitely cause vomiting and typically goes along with hyperthyroidism (as well as just being a geriatric cat). Always a good idea to check liver values as well, as liver disease is a common problem in older cats too.
So since your cat is hyperthyroid the first step to diagnosing causes of vomiting is running full blood work - complete blood count, chemistry panel, and urinalysis - to look for some of the things I mentioned above. If nothing turns up, imaging with x-rays or ultrasound or both will likely provide a lot more information. Good luck.
A. I don't typically think of coughing as being associated with hyperthyrodism. Are you sure your cat isn't vomiting, or trying to vomit? Hyperthyroid cats will definitely vomit when their disease is uncontrolled, which I'm assuming is the case with your cat since the diagnosis is recent (and it take a while to get them regulated on medication).
If the coughing continues past this one episode, I would definitely call your vet and discuss the coughing. Hyperthyroidism can cause secondary heart failure, and while cats with heart failure typically don't cough (like dogs do), anything's possible, so it needs to be discussed.
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