Paronychia: This is the most likely diagnosis given the patient's history of biting his nails, the recent hangnail, and the current presentation of redness, swelling, and pain around the nail. Paronychia is an infection of the skin around the nail and is commonly seen in individuals who bite or pick at their nails.

Cellulitis: This is an infection of the skin and subcutaneous tissues and could be a consideration given the redness, swelling, and pain. However, it typically involves a larger area of skin than paronychia.
Felon: A felon is an infection of the fingertip pad and could be considered if the infection is more localized to the pad of the finger. However, the presence of a hangnail and the location around the nail make paronychia more likely.
Herpetic Whitlow: This is a viral infection of the finger caused by herpes simplex virus. It could be considered if there are vesicles or a history of exposure to herpes simplex. However, the absence of these findings makes it less likely.

Osteomyelitis: Although less likely, osteomyelitis (an infection of the bone) could occur if the infection spreads from the soft tissues to the bone. This would be a serious complication and could lead to significant morbidity if not promptly treated.
Septic Arthritis: If the infection spreads to the joint, it could cause septic arthritis, which is a medical emergency requiring prompt treatment.
Endocarditis: In patients with certain risk factors (e.g., pre-existing heart conditions), an infection like paronychia could potentially lead to endocarditis, an infection of the heart valves.

Erysipelas: A type of skin infection that involves the upper dermis, typically caused by group A beta-hemolytic streptococci. It's less common than cellulitis and usually has a distinct border.
Sporotrichosis: A fungal infection that can occur after a skin injury, such as a hangnail, especially if the patient has been exposed to soil or plants. It's rare and typically presents with a nodular lesion that progresses along the lymphatic channels.