**Core Concept**
The child's presentation of a non-blanching rash, proteinuria, and hematuria is suggestive of Henoch-Schönlein purpura (HSP), a form of vasculitis that affects small blood vessels, particularly the capillaries in the skin and kidneys. This condition is characterized by deposition of IgA immune complexes.

**Why the Correct Answer is Right**
The characteristic finding in HSP is the deposition of IgA immune complexes in the affected tissues, including the skin and kidneys. This leads to inflammation and damage to the small blood vessels. In the kidneys, this can result in glomerulonephritis, characterized by proliferation of the glomerular mesangial cells and deposition of IgA immune complexes. The presence of proteinuria and hematuria is indicative of kidney involvement.

**Why Each Wrong Option is Incorrect**
**Option A:** Focal segmental glomerulosclerosis (FSGS) is a different glomerular disease characterized by scarring of the glomeruli, which is not typically associated with HSP.
**Option B:** Minimal change disease is another glomerular disease, but it is characterized by normal glomeruli on light microscopy and is not typically associated with HSP.
**Option C:** Membranous nephropathy is a different glomerular disease characterized by thickening of the glomerular basement membrane due to immune complex deposition, but it is not typically associated with HSP.

**Clinical Pearl / High-Yield Fact**
HSP is a classic example of a systemic vasculitis that affects small blood vessels, particularly in children. It is characterized by the deposition of IgA immune complexes and can involve multiple organs, including the skin, kidneys, and gastrointestinal tract.

**Correct Answer:** C.