wow sounds like a deco dive for sure if you are down that long. How many tanks do you use on avarage for this dive? do you mainly use EANx? how long on avarage are your dives? and how far can an intro to cave diver go in a cave like the devils ear/eye/little devil? I ask because i am looking into takeing the... I wanna say its 4 or 6 day Cavern and intro to cave class combined.
As Rox stated, intro is limited to 1/3 of a single tank or 1/6 of doubles, and mainline only. One thing he did not mention, is that basic/intro/cave1 all prohibit decompression during cave dives.
When Litehedded and I were both basic, we managed to squeak out a dive around 900ft back in Devils', and it took about a total of 60 minutes there and back (5min of which was minimum deco). However, my first dive in devils under intro limits my team didn't make it but maybe 200ft before hitting our turn pressure, and about a 15minute dive. At peacock springs during my basic cave class, we made it 900+ft up the peanut line. After calming down some a few weeks later, we made it 1200ft. The more you dive, the better your teamwork, the better your trim, and the more streamlined your gear.....the further you'll get.
As for your deco question, PFCAJ turned me on to the following set of tables that work GREAT for basic cave dives when you need that extra repetitive dive time (be sure to read the 2nd link on avg depth). Most basic cave dives can be done with minimum deco from my experience. Remember, most caves are *not* square profiles. In Ginnie as a new diver, your first 15-20min of penetration is all shallower than 60ft....IIRC I don't think you get past 60ft deep until after the keyhole on the map.
DIR-diver.com - Using the min deco table
DIR-diver.com - Average depth for deco?
You asked nitrox vs air, I'd avoid air, 32% Nitrox has a MOD that you can do most any cave in the area, and IMO the minor cost savings over nitrox isn't worth adding a decompression ceiling to every dive.
After you've practiced your technique in the flow, to get to the white room, you'll use about 600psi, and Hill 400 takes about 300-400psi (psi for double 104/108/130s). Neither are deco dives.
Here's some of the better cave videos you might enjoy-
Hill 400 from the bats -
http://www.youtube.com/watch?v=N0wVA721Y-I
Scootering Mainline Ginnie-
http://www.scubadivingtube.com/play.php?vid=497
Andrew Ainslie's Page (lots of good cave video)-
http://andrewainslie.com/page3.html
I couldn't find any video of the white room, but on this page, the 3rd and 4th image are the white room and the entrance to it
http://www.travel-dive.com/why-ginnie-and-i-are-like-peas-and-carrots
A word of advice, don't fall for the trap of thinking cave diving is all about going really far back....it's about seeing things. The secret to get really far into a cave is to stop caring how far you get, slow down and enjoy the scenery. Saturday, we swam to double domes (1800ft back), and then I had just as much fun doing the lips bypass (300-400ft back) on the very next dive. Distance is just a number, not a measure of fun!