Not without the ok of your veterinarian.
If your puppy has a stomach ulcer, then Yes!
This is possible, depending upon what the shot was. If it was a vaccination, the puppy is probably not vomiting from the shot itself, although I would keep an eye on the puppy during and after the next round of vaccinations. More likely, your puppy is vomiting from motion sickness and stress from riding in the car to and from the vet's office.
induce vomiting somehow or go to the hospital
Yes. Aspirin is a known potential irritant to the gastric mucosal lining. If aspirin has been recommended/prescribed by your physician, it should be taken with food, not on an empty stomach.
Aspirin dissolves in warm water. If cold water is used, that will give a maximum yield.
I am not sure but don't give your dog an aspirin just to be safe.
thats means its vomiting (simply classic)
Yes, but avoid preparations with aspirin in them, as Lortabs contain aspirin.
Reye's syndrome
No you can't give it aspirin because it would choke wait until 1 year or 2 and then you could give it aspirin.
a puppy that needs help
If your dog has ingested 81 mg of aspirin, it's important to contact your veterinarian or an emergency animal poison hotline immediately. Aspirin can be toxic to dogs, and the severity depends on your dog's size and health status. Monitor for symptoms such as vomiting, lethargy, or changes in behavior while you seek professional advice. Do not induce vomiting unless instructed by a veterinarian.