How to run LED lights off wild DC from bike powered generator
For Problem 1:
A string of LEDs hooked up in series with a current regulator like SuperTex CL220 would do the trick: The component is a simple a 2-terminal device (like a diode) and needs no additional components or configuration. It allows 20 mA (+/-10%)  current to pass through as long as there is sufficient voltage headroom: 5 volts above the total forward voltage of the LEDs is sufficient. This current regulation is stable up to 160 volts, enough for your purposes.

Note that LEDs are current-dependent rather than voltage dependent devices. They glow at essentially the same brightness as long as the current is constant. 

For your application, if the LEDs need to start glowing from around 60 Volts, half the  120 Volts your students generate, then a string of 25-30 standard 5-mm red LEDs would be optimal. They would glow without intensity change till your maximum voltage. 

Too many LEDs = they won't glow till a higher voltage.
Too few LEDs = the CL220 device would overheat in dissipating the surplus voltage.



For Problem 2:
The extent of energy storage required to keep a string of around 25 LEDs (from above section) glowing for even half a minute, is pretty high. Capacitors would not be the way to go, unless you have access to big power-line capacitors through surplus channels.


Your capacitor bank would need to provide 20 mA at a minimum of 60 volts (again from above section), for "a few" seconds. 
Capacitor needs to be rated for a voltage higher than the highest the generator could conceivably generate.
Though "supercapacitor" is a popular term these days, typical supercaps are rated for 5.5 Volts or 12 Volts, not hundreds of Volts.
Adding in buck/boost generator trickery to make this work would result in complexity far beyond using a battery and off-the-shelf charger.


I hope this helped.