Final answer:
The diagnosis for the presented case with jaw pain, fever, and swelling near a tooth is likely a dental abscess, characterized by a localized infection resulting in pus accumulation.
Step-by-step explanation:
The most likely diagnosis for a 47-year-old male presenting with severe pain in the lower jaw, fever, facial swelling, a foul breath odor, and an erythematous, fluctuant, tender mass near the lower third molar is dental abscess. The constant, throbbing pain that worsens when lying down is characteristic of an abscess, where pus accumulates due to a bacterial infection. This is in contrast to mumps, which typically involves swelling of the parotid glands and would not present as a localized fluctuant mass near a specific tooth.