Hello and thank you for your questions.
The tooth that is broken off at the gum line is likely abscessed. An Abscess is a localized infection entrapped in soft tissue
and bone that is full of pus. You may have a separate sinus
issue or because the roots of the teeth are in very intimate approximation with the maxillary sinus, the infection may have spread into your sinus cavity.
Infections get into your blood stream by spreading into the local microvasculature which then caries the bacteria to your heart which pumps it to other parts of your body and gives you an overall feeling of weakness and sickness (malaise
The little white bump on your tooth is where the infection has eaten its way through the bone and is now building up pressure below the tissue. It is basically becoming the head of a pimple in your mouth.
The risk of the infection is that the maxillary sinus is close to your brian and you could get the infection to spread to your brain which can lead to death. You have antibiotics but this is only a bandaid. The best way to address this issue is to go to your dentist
or oral surgeon, get an incision and drainage
of the infection and have the tooth extracted to remove the infection. Do this ASAP because infections can spread rapidly and should not be messed with. The chances of it getting into your brain are very low but because there is a risk involved, I would get the issue addressed.
Hope you feel better!