DarkSide-50 is the current phase of the DarkSide direct dark matter search program, operating underground at the Laboratori Nazionali del Gran Sasso in Italy. The detector is a dual-phase argon Time Projection Chamber (TPC), designed for direct detection of Weakly Interacting Massive Particles (WIMPs), and housed within a veto system of liquid scintillator and water Cherenkov detectors. Since switching to a target of low radioactivity argon extracted from underground sources in April 2015, the background is no longer dominated by naturally occurring 39Ar. However, alpha backgrounds from radon and its daughters remain, both from the liquid argon bulk and internal detector surfaces. In some cases, the alpha events are a dangerous background in need of mitigation. In others, the alpha events can be used as a tool to deepen our understanding of our detector. We present here several analyses of alpha events in DarkSide-50. An analysis of radon-polonium coincidences is performed, yielding measurements of ion mobility and charged daughter fraction in liquid argon. These coincidences are also used to place upper limits on XY-resolution and liquid argon motion in the detector. A method for simulating the degraded energy spectra from 210Po events traveling through surface materials is presented and compared with data from DarkSide-50. Lastly, a pileup background created by the accidental coincidence of scintillation-only 210Po surface events with random ionization-like pulses is thoroughly discussed. The estimation of this background and its mitigation informed the recent WIMP search analysis performed with 532 live-days of DarkSide-50 data.