Controlling the stability of nanostructures is an important fundamental issue in nanoscience. A key problem in this respect is the random stochastic interface dynamics associated with equilibrium nanometer scale thermal fluctuations, which can be understood by applying first--passage statistical concepts, such as the temporal persistence [survival] probability $P(t)$ [$S(t)$]. $P(t)$ [$S(t)$] measures the probability of the step height $not$ returning to the original [average] value within a given time interval. In this study, we measure the persistence exponent $\theta$ associated with the power--law decay of $P(t)$ at large time scale and also the survival time scale, $\tau_s$, describing the exponential decay of $S(t)$ for several nonequilibrium surface growth processes and fluctuating steps on solid surfaces. We show that for growth models in the Molecular Beam Epitaxy universality class, the nonlinearity of the underlying equation is clearly reflected in the difference between the measured values of the positive and negative persistence exponents. We also find that the survival time scale provides useful information about the physical mechanism underlying step fluctuations. An exact relation between long-time behaviors of the survival probability and the autocorrelation function is established and numerically verified. The dependencies of the persistence and survival probabilities on the system size and the sampling time are shown to be described by simple scaling forms. We introduce the generalizations of the persistence and survival probabilities for surface fluctuations. Our measurements of the persistent large deviations probability and associated family of exponents show good agreement with the corresponding quantities measured experimentally using scanning tunneling microscopy dynamical images of step fluctuations. The spatial persistence and survival probabilities are also theoretically analyzed. Finally, we investigate the connection between the generalizations of the spatial and temporal persistence probabilities.