N2O production vs air distillation
After producing nitrogen by distilling air, how is that nitrogen used with the oxygen to yield nitrous oxide ($\ce{N2O}$)?


It's not clear to me that $\ce{N2O}$ is produced from $\ce{N2}$: Rather, it appears that commercial quantities of $\ce{N2O}$ are produced by thermally decomposing ammonium nitrate:
$$\ce{NH4NO3(s) -&gt; N2O(g) + 2H2O(g)}$$
as described (at least) by this manufacturer and supplier of nitrous oxide gas plants. See the comprehensive Wikipedia entry for more information.


  I see $\ce{N2O}$ available in different purities from various suppliers, but how is that different from just distilling air?


I'm not sure what you mean here.


  When distilling nitrogen from air, once both gasses have reached the right temp/pressure to become a liquid, what is preventing it from becoming $\ce{N2O}$ in a liquid form instead of one floating on top of the other?


Nitrogen is liquid in the range 63 to 77 K; for oxygen, it's 54 to 90 K. Thus, the temperature range for which both are liquids is bounded by the range on nitrogen, or roughly between 63 and 77 K.

The temperature at which nitrous oxide solidifies (182 K per Wikipedia) is much higher than the temperature range at which oxygen and nitrogen are in the liquid phase, and thus it would have come out of solution as a solid long before the requisite temperature was reached. Thus, we would not expect to see liquid $\ce{N2O}$ co-existing with liquid oxygen and nitrogen.

Note bene: I am using the pressure range of 5 to 10 bars as cited here in assuming that liquid and solid phases of the above moieties do not coexist during industrial gas production at these relatively low pressures.