While you can't capture the full grandeur of the big room with flash, that's just one shot out of many. Most of the interesting formations are within flash range.
I recommend using both flash and long exposure shots.
The natural colors of the stone are spectacular, and only a flash shot will give you the full impact. Far from being "flat"; the complexity of the stonework does a good job of diffusing even harsh flash lighting. The biggest problem with on camera flash is uneven lighting in shots with significant foreground/background elements.
The problems with timed shots are many; Light tripods are not really adequately steady for multi-second exposures, and heavy tripods are ..um.. heavy. *All* the lighting in the cave is artificial, incandesent, and frequenly colored with simple gels. To the human eye, the effect isn't bad, but on film, blue and pink rocks just don't look right. Plus the poor spectral quality of the light doesn't reveal the stones' true colors. I also used a tungsten filter some of the time, in an attempt to bring the color temperature closer to normal. It helped, but made exposure times very long.
the overall light level is enough that camera metering systems are fairly trustworthy; but bracket anyway.
not a problem except in the tourist-heavy areas near the concession stand.
Take the guided tours to the areas not open to the public. Take the "walk in" trail from the natural entrance. Stick around for the sunset exodus of the bats.
I spent two days. Not enough, but unless your companions are equally enthralled and also into photography, don't try for any more. All that fooling around with tripods in the dark takes a lot of time.
Shoot lots of film, use different lighting techniques, bracket. (Unless you live in the neighborhood, you won't be back soon.)