What would be a complete QC circuit? In classical computing NAND is a complete set (functionally complete) of binary operations, namely any Boolean circuit can be expressed using NAND gates. 

Is there an equivalent for quantum computing circuit? It should be possible imho to have such a complete set because any rotation ob Bloch sphere is reducible to Euler angles based rotation matrices: $R=X(\alpha)Y(\beta)Z(\gamma)$.
Are there any universal quantum gates?

Not for single gates.

{NAND} may also be represented as two gates {AND, NOT}, and remains a minimally functionally complete set of classical gates.

The Hadamard gate and Toffoli gate are one minimal set of universal quantum gates.