Infectious Mononucleosis (Mono): This condition, caused by Epstein-Barr virus (EBV), can present with sore throat, cough, and systemic symptoms such as fever and fatigue. The joint pain and swelling could be related to the immune response or associated conditions like arthritis. Bilateral lower leg swelling could be due to secondary infections or inflammation.

Viral Syndrome: A broad term that encompasses various viral infections that could cause the combination of symptoms described, including upper respiratory symptoms (cough, sore throat) and systemic symptoms (joint pain, swelling).
Streptococcal Infection: Streptococcal pharyngitis can lead to sore throat and cough, while post-streptococcal glomerulonephritis or rheumatic fever could explain the joint pain and swelling, and potentially the leg swelling if there's associated nephrotic syndrome.
Influenza: Especially during flu season, influenza can cause systemic symptoms including joint pain, cough, and sore throat. Secondary bacterial infections could lead to more localized symptoms like leg swelling.

Deep Vein Thrombosis (DVT): Although less common in young adults without risk factors, DVT can cause unilateral or bilateral leg swelling and should be considered, especially if there's a history of recent immobility, trauma, or family history of clotting disorders.
Lymphoma: Certain types of lymphoma can present with systemic symptoms including fever, night sweats, weight loss, and sore throat. Joint pain and swelling could be due to bone marrow involvement or associated autoimmune phenomena.
Endocarditis: Infective endocarditis can cause systemic symptoms including joint pain, fever, and potentially leg swelling due to embolic phenomena. A new heart murmur or other cardiac signs would increase suspicion.
Sepsis: Bacterial sepsis can cause a wide range of symptoms including fever, chills, joint pain, and swelling, especially if there's an associated infection like pneumonia (which could explain the cough).

Kawasaki Disease: Although more common in children, Kawasaki disease can occur in young adults and presents with fever, rash, conjunctivitis, and cervical lymphadenopathy, along with potential cardiac and joint involvement.
Sarcoidosis: A systemic granulomatous disease that can cause a wide range of symptoms including joint pain, swelling, and respiratory symptoms like cough. It's less common but should be considered in the differential for unexplained multi-system disease.
Wegener's Granulomatosis (Granulomatosis with Polyangiitis): A form of vasculitis that can cause respiratory symptoms (cough), joint pain, and swelling, along with renal involvement that could lead to leg swelling due to nephrotic syndrome.