Quasi-phase-matching (QPM) has enriched the capacity of parametric down-conversion (PDC) in generating biphotons for many fundamental tests and advanced applications. However, it is not clear how the nonidealities and randomness in the QPM grating of a parametric down-converter may affect the quantum properties of the biphotons. This paper intends to provide insights into the interplay between PDC and nonideal or random QPM structures. Using a periodically poled nonlinear crystal with short periodicity, we conduct experimental and theoretical studies of PDC subject to nonideal duty cycle and random errors in domain lengths. We report the observation of biphotons emerging through noncritical birefringent-phasematching, which is impossible to occur in PDC with an ideal QPM grating, and a biphoton spectrum determined by the details of nonidealities and randomness. We also observed QPM biphotons with a diminished strength. These features are both confirmed by our theory. Our work provides new perspectives for biphoton engineering with QPM.