Can a two variable Harmonic function f(x,y) be zero on a curve with a cusp?
No. A two variable harmonic function is the real part of an analytic function. Near a zero, an analytic function behaves like a power of $zz_0$. 


Perhaps study this one: $$ f(x,y) = \frac{2 \sqrt{x^{2} + y^{2}}  \sqrt{2 \sqrt{x^{2} + y^{2}} + 2 x}}{2 \sqrt{x^{2} + y^{2}}  2 \sqrt{2 \sqrt{x^{2} + y^{2}} + 2 x} + 2} $$ which vanishes on a cardioid with a cusp at the origin. Harmonic except possibly at the one point $(0,0)$. 

