This study considers the unconditional smooth Rényi entropy proposed by Renner and Wolf ASIACRYPT, 2005, the smooth conditional Rényi entropy proposed by Kuzuoka IEEE Trans. Inf. Th., 66(3), 1674-1690, 2020, and a novel quantity which we term the conditional smooth -⋆ entropy. The latter two quantities can be specialized to the first in the absence of side-information. We explore the operational roles of these smooth Rényi entropies by establishing one-shot coding theorems for several information-theoretic problems, including Campbell's source coding problem, the Arıkan-Massey guessing problem, and the Bunte-Lapidoth task encoding problem. We consider these problems in cases where the errors are non-vanishing and for each problem, we consider two error formalisms: the average and maximum error criteria, where the averaging and maximization are taken with respect to the side-information. Using the one-shot coding theorems, we conclude that Kuzuoka's smooth conditional Rényi entropy and the conditional smooth-⋆ entropy are the solutions to the problems involving the average and maximum error criteria, respectively. Furthermore, we examine asymptotic expansions of these entropies when the underlying source with its side-information is stationary and memoryless. Applying our asymptotic expansions to the one-shot coding theorems, we derive various fundamental limits for these problems. We show that, under non-degenerate settings, the first-order fundamental limits differ under the average and maximum error criteria. This is in contrast to a different but related setting considered by the present authors IEEE Trans. Inf. Th., 66(12), 7565-7587, 2020, for variable-length conditional source coding allowing errors, in which the first-order terms are identical but the second-order terms are different under these error criteria.