If serious, yes! A serious and acute dental infection, especially one threatening to block a person's ability to breathe may very well require a dose of IV antibiotics to get the infection under control until the offending tooth (teeth) can be treated.
Antibiotic injection. There are injections that go under the gums that are a low dose minocycline hcl called Arestin that do not treat tooth infections but may help a little with gum infections if used together with deep cleaning or some inflammation, not all dentists use it. Consult w yours if that is what your referring to.
Can be but. Most of the time this occurs it is in a hospital/er setting and is only a temp fix and you should see a dental provider for care. If the tooth is not taken care of then the antibiotic injection will not be sufficient to resolve the issue completely. If it is a tooth infection you are going to need a root canal or extraction guaranteed.
Antibiotics. Yes, possibly if the infection is severe. However, the antibiotic is more often prescribed to be taken by mouth as most tooth and gum infections are mild. Penicillins are the drug of choice. This family of antibiotics includes amoxicillin, ampicillin. Etc.