Incorrect. Dysphagia is a difficulty swallowing, not directly related to a symptom of drooling.

Correct! The marginal mandibular nerve is the branch of the facial nerve responsible for control of the muscles of facial expression along the ramus of the mandible. This region includes the orbicularis oris region in the lower lip, depressor labii inferioris and depressor anguli inferioris. Palsy of this nerve would impact function of the lower lip leading to possible drooling.

Inflammation of the submandibular region could cause inflammatory mediated neuropathy of the nerve as it courses along the superficial tissues of the mandible, particularly as the patient has submandibular cellulitis the subcutaneous tissue in this region is inflamed.

Incorrect. The mental nerve emerges from the mental foramen near the chin (the mentum). Additionally, this nerve is a sensory nerve and would affect sensation at the chin and not motor control of the lip responsible for the patient drooling.

Incorrect. The mandibular nerve is the trigeminal V3 nerve branch, responsible for sensation along the lower face (jawline) and motor control of the muscles of mastication.

Neuropathy would result in sensory deficits and deficits in chewing, but motor control of the lips themselves is via another nerve.

Incorrect. The zygomatic nerve is a branch of the facial nerve coursing inferior to the eye. It covers territory principally of the lower eye with some coverage of the upper lip (also covered by the buccal nerve running parallel and inferior to the zygomatic branch). 

Neuropathy could cause eye closure difficulty and minor upper lip control. Some drooling might occur, but since the nerve is quite distant from the submandibular cellulitis region a subcutaneous tissue inflammatory neuropathy of the zygomatic branch is unlikely.

Incorrect. Edema of the tongue would result in dysphagia, but not directly drooling as the muscles responsible for the lips are not shared with the tongue.