What will be the output/discharge current of a capacitor?
but I am not sure what will be the output voltage and current of the
  capacitor would be to the boost converter


voltage : output voltage will be the same ass input voltage(2V)

current : i = C(dv/dt) = 25(2/time) Will depend upon the load.

Energy stored in capacitor=  C * V^2 / 2 = 25*2^ = 100 J or Watt(as solar panel is giving 2V )

2V 500mA solar panel will take 100 sec to full charge the 25F 2V capacitor(depends on the sun light and solar panel output)


  Another set up would be to connect 4 25F 2.7V capacitors in series and
  then in parallel to a 1.2 watt solar panel, what would be output V and
  I in that case


4 capacitor in series + solar panel 1.2W in parallel? need voltage and current (if solar panel voltage is less than 8V nothing will happen)

voltage : 4 * input voltage

current : i = C(dv/dt) = C(2/time) Will depend upon the load.

1/C = 1/c1 + 1/c2 + 1/c3 + 1/c4 put C value in above energy capacitor formula.