Rheumatoid Arthritis (RA): The presence of bilateral leg swelling, joint pain, and swelling, along with nodules in the upper arms (possibly rheumatoid nodules), are classic symptoms of RA. The initial cough and fever could be related to an infectious trigger or an extra-articular manifestation of RA.

Lyme Disease: Given the initial cough and fever that resolved with antibiotics, and the subsequent development of joint pain and swelling, Lyme disease is a consideration, especially if the patient has been exposed to ticks. The nodules could be related to Lyme disease, although they are more commonly associated with RA.
Reactive Arthritis: This condition can occur after a bacterial infection, which might explain the initial cough and fever that resolved with antibiotics. It typically presents with joint pain and swelling, and sometimes skin lesions or nodules.
Sarcoidosis: This systemic disease can cause joint pain and swelling, skin nodules, and lung symptoms like cough. The resolution of initial symptoms with antibiotics might not be directly related to sarcoidosis, but it's a possibility if the antibiotics coincidentally improved symptoms.

Infective Endocarditis: Although less likely given the resolution of fever with antibiotics, infective endocarditis can cause joint pain, fever, and sometimes skin manifestations like Janeway lesions or Osler's nodes. Missing this diagnosis could be fatal.
Lymphoma: Certain types of lymphoma can cause systemic symptoms like fever, cough, and joint pain, along with skin nodules. It's crucial to consider lymphoma to avoid missing a potentially life-threatening condition.
Vasculitis (e.g., Polyarteritis Nodosa, Granulomatosis with Polyangiitis): These conditions can present with a wide range of symptoms including joint pain, skin nodules, and systemic symptoms like fever and cough. They are critical to diagnose early due to their potential for serious complications.

Relapsing Polychondritis: A rare autoimmune disorder that can cause joint pain, skin nodules, and systemic symptoms. It's less likely but should be considered in the differential due to its unique presentation.
Multicentric Reticulohistiocytosis: A rare condition characterized by skin nodules and joint pain, which could fit some of the patient's symptoms, although it's much less common than other diagnoses listed.