We study the typical-code (quenched) behavior of the false-alarm (FA) and missed-detection (MD) error exponents of the Neyman-Pearson test associated with soft covering, complementing the average-code (annealed) analysis that has been carried out in a companion paper [1]. We prove that, as the block-length tends to infinity, for almost every randomly selected fixed-composition codebook, the negative normalized logarithms of both error probabilities converge to their respective average-code exponents. In other words, the error exponents are self-averaging. We then extend the scope and study a mismatched likelihood ratio test that assumes the wrong channel model. Here, we derive the mismatched error exponents, show that self-averaging persists under mismatch, and characterize the degradation. In particular, we characterize the coding rate beyond which the two kinds of error exponents cannot be positive at the same time, which in the matched case, is given by the channel input-output mutual information rate.