Light sterile neutrinos might mix with the active ones and be copiously produced in the early Universe. In the present paper, a detailed multi-flavor analysis of sterile neutrino production is performed. Making some justified approximations allows us to consider not only neutrino interactions with the primeval medium and neutrino coherence breaking effects, but also oscillation effects arising from the presence of three light (mostly-active) neutrino states mixed with two heavier (mostly-sterile) states. First, we emphasize the underlying physics via an analytical description of sterile neutrino abundances that is valid for cases with small mixing between active and sterile neutrinos. Then, we study in detail the phenomenology of (3+2) sterile neutrino models in light of short-baseline oscillation data, including the LSND and MiniBooNE results. Finally, by using the information provided by this analysis, we obtain the expected sterile neutrino cosmological abundances and then contrast them with the most recent available data from Cosmic Microwave Background and Large Scale Structure observations. We conclude that (3+2) models are significantly more disfavored by the internal inconsistencies between sterile neutrino interpretations of appearance and disappearance short-baseline data themselves, rather than by the used cosmological data.