We study stochastic integrals driven by a general subordinator and establish a zero-one law for the finiteness of the resulting integral as well as moment estimates. As an application, we use these results to obtain structural properties of SPDEs driven by multiplicative pure jump noise, which include (1) a maximal inequality for a multiplicative stochastic convolution Zt, (2) a small ball probability of Zt, (3) the existence of invariant measures and accessibility to zero of SPDEs, and (4) a Galerkin approximation of solutions to SPDEs.