Its better to remove the tooth that has a problem than to leave it in. If the infection hits the root of the tooth, the dog will also have diareah as well as the above- .. they will figure it out. You will have to soak the food or as directed by the vet at first until the gums heal but really- healthy teeth are better..
It is strange a vet would suggest anethesia at age 10 though.. hmmm.