Here we study fluctuations, scaling, and universality in a variety of first-passage scenarios. First, we explore fluctuations in fixation times in evolutionary dynamics. We compute the fixation-time distribution for several models of evolution and determine how the shape of the distribution depends on the fitness advantage provided by a genetic mutation. Our results reveal an interesting dichotomy: for neutral mutations the distribution is highly-skewed, while for non-neutral mutations, two particular distributions arise. In the latter case, depending on population structure the fixation-time distribution is either a Gaussian or the (moderately skewed) Gumbel distribution. Next, we show that the Gaussian and Gumbel distributions are universal; they arise generically across a variety of stochastic models of evolution, ecology, epidemiology, and chemical reactions. The distinguishing feature is the decay of the stochastic transition rates near the absorbing state: lack of decay leads to Gaussian distributions, while linear decay leads to Gumbel distributions. Distributions resulting from other power-law decays in the transition rates are also classified. Finally, we formulate a renormalization group approach and scaling theory for barrier crossing phenomena near a noisy saddle-node bifurcation, where the barrier vanishes. We derive the universal scaling behavior and corrections to scaling for the mean barrier escape time in overdamped systems with arbitrary barrier height. We also develop an accurate approximation for the fluctuations in escape times, capturing the full distribution of barrier escape times at any barrier height. This critical theory draws links between barrier crossing in chemistry, the renormalization group, and bifurcation theory.