How Would A Vast Mixture (Like an Ocean) of Ammonia-Water Behave in Terms of Evaporation and Freezing?
Freezing


  I've heard that the freezing and boiling points of the mixed liquid would exist on some continuum between the freezing and boiling points of its constituent liquids (ammonia and water), varying depending on the exact mix ratio and of course pressure. Is that correct?


For freezing points, that is incorrect. Pure water freezes at 273 K and pure NaCl (table salt) "freezes" at 1074 K. However, if I dissolve NaCl in water, the freezing point of water will be lowered. In you water/ammonia scenario, water will freeze as pure water, but the freezing point will be lower. Ice formation would increase the concentration of ammonia in solution.

Evaporation

Water and ammonia mix in the liquid state, and everything mixes in the gas state. Evaporation (and condensation) depend on the ratio of components in the liquid state, the partial pressure of components in the gas state (or ratio of components and total pressure), and the temperature. At equilibrium at earth-like temperatures, there would be more ammonia than water in the atmosphere (the boiling point of ammonia is much lower than that of water, so the vapor pressure of ammonia is higher than that of water). A sudden drop in temperature would make it rain ammonia, mixed with a bit of water. So condensation and precipitation would increase the concentration of ammonia in the liquid state, and a sunny day would lower it.


  If the later behavior is correct I could imagine an ocean of water-ammonia varying substantially in its water/ammonia ratio from pole to equator thanks to the greater heat near the equator and the ammonia molecules evaporating.


You are making a lot of assumptions here: that the exoplanet rotates fairly quickly around its own axis, that its axis is roughly perpendicular to the star, and that the star is close enough to heat up the planet.  


  What I'd like to most wrap my head around is whether the water-ammonia mix would behave as a single uniform liquid, freezing and evaporating at a rate determined by the average of the mix's freezing/boiling point, or whether the ammonia molecules would start to evaporate before the water molecules, and the water molecules start to form ice before the ammonia molecules.


No, they would do different things. Pure water at terrestial atmospheric pressure freezes at 0 $^\circ$C, while ammonia already boils at -33 $^\circ$C. Mixing them will lower the freezing point of water and increase the boiling point of ammonia, so there is no temperature where you would separate them completed by freezing all water and boiling off all ammonia. On the other hand, they don't freeze or boil in lock step. If you lower the temperature, water will form ice and the ammonia concentration in the liquid will increase. If you increase the temperature, more ammonia than water will turn into gas (evaporating or boiling), so the ammonia concentration in the liquid will decrease.