Increase low-frequency gain of a cascode amplifier
If I were to build this circuit using an aluminum electrolytic
  capacitor for C2, will I run into ESR problems at high frequencies?


Not only ESR but ESL (inductance) will cause a resonant frequency and a gain peaking somewhere in the mid kHz to tens of kHz range and above this, gain will gradually reduce as the inductor becomes more dominant.


  How can I reduce the value of the emitter bypass capacitor (C2) while
  maintaining the desired frequency response and passband gain?


Difficult given that you need a gain of ten and that you don't want the output impedance to rise much more than 680 ohms - I would consider using ceramic 22 uF capacitors and parallel them up to give you 220 uF. Again, you have to watch out for resonant frequency problems and you may need to add some 100 nF capacitors across those 22 uF caps to get a fairly flat response.

Try looking up a few and adding the parasitic components to your model. So, to answer your first question: -


  Does my simulation provide a good approximation of the desired
  behavior? Recall that the goal is to amplify white noise between 10Hz
  and 10MHz.


No, but this is easily remedied by adding parasitic components.