You cannot contract HPV in this manner unless you are the one who actually handled the dirty tampon. It is a very rare to unlikely occurrence that any contact not sexual can transmit the disease. If you remain concerned or did handle the dirty tampon without cleaning your hands or using gloved hands or other means to protect yourself then I would suggest contacting your family doctor. Again the risk is very low that you can contract the virus in this manner.
As for your dog. This is one of many conditions that are not transmissible from human to dog or vice versa.
Please let me know if further assistance is needed.
You are very welcome.
It isn't necessary to wash out the dogs mouth and yes he can lick you. The virus does not live long outside of the body and inside the mouth of a dog it cannot survive because a dog is not the host.