Henoch-Schönlein Purpura (HSP): This condition is characterized by the deposition of IgA immune complexes, leading to vasculitis. It classically presents with palpable purpura, abdominal pain, arthralgias, and renal involvement. The presence of bilateral purpura on the calves that does not recede under pressure (non-blanching) is highly suggestive of HSP. Dry cough could be an atypical presentation or an associated symptom due to potential pulmonary involvement.

Infectious Endocarditis: Although less common, infectious endocarditis can present with purpura due to septic emboli or immune complex deposition. The dry cough might be related to pulmonary emboli or heart failure symptoms.
Vasculitis (e.g., ANCA-associated vasculitis): Conditions like granulomatosis with polyangiitis (GPA) or microscopic polyangiitis (MPA) can present with purpura and respiratory symptoms, including dry cough, due to the involvement of small vessels.
Meningococcemia: This is an infection caused by Neisseria meningitidis, which can present with purpura and might have respiratory symptoms, although the latter is less common.

Disseminated Intravascular Coagulation (DIC): Although less likely, DIC can cause purpura and could be associated with a dry cough if there's pulmonary involvement. It's crucial to consider DIC due to its high mortality rate if left untreated.
Thrombotic Thrombocytopenic Purpura (TTP): TTP is characterized by thrombocytopenia, microangiopathic hemolytic anemia, renal failure, neurological symptoms, and fever. Purpura can be a presenting feature, and while less common, respiratory symptoms like dry cough can occur.
Sepsis: Sepsis from any cause can lead to purpura and respiratory symptoms, including dry cough, especially if there's an associated pneumonia or acute respiratory distress syndrome (ARDS).

Ehlers-Danlos Syndrome with associated vascular fragility: This could lead to easy bruising and purpura, and while less directly linked to dry cough, the syndrome's vascular complications might indirectly cause respiratory symptoms.
Scurvy: Caused by vitamin C deficiency, scurvy can lead to perifollicular petechiae or purpura and might have associated respiratory symptoms, although this is rare and typically seen in the context of severe malnutrition.
Amyloidosis: Certain types of amyloidosis can cause purpura and might be associated with respiratory symptoms, including dry cough, especially if there's cardiac or pulmonary involvement.