cryptographically good random elliptic curves?
No, not all curves are created equal. For one thing, the number of points on an elliptic curve defined over $\mathbf{F}_p$ varies in the interval $[p+1-2\sqrt{p},p+1+2\sqrt{p}]$ (Hasse's Inequality). There are also weak instances which must be avoided, the most catastrophic being when the number of points on the curve is exactly equal to $p$ (anomalous curves) or to $p+1$ (supersingular curves).