You have an array of LED pins. Why don't you use it?
Create functions for the various conditions you want the LEDs to represent (only LED 1 on, only 2 on, etc.) and call those functions.
am trying to get the LED's to light up randomly
Where? The code you posted does specific things in response to specific button presses. Randomly turning on LEDs is possible.
Should there be just one LED on at a time, or can there be multiple LEDs lit?
then stay on for a given amount of time
Look at the Blink Without Delay example.
then be turned off by its corresponding push-button.
Very easy.
I also want to be to record if the button isnt pushed in the given time
The Blink Without Delay example will get you going in the right direction.
and if the 'player/participant' fails 3 times, then its game-over.
Since the Arduino is always doing something, this gets a bit trickier. Counting the number of fails is easy. Stopping turning on the LEDs is easy. Doing nothing is easy.
Resuming doing something will probably require another switch or creative use of the switches that you already have (think ctrl-alt-delete).