I've seen only one cut-down tire... I don't think Max or Dave have spares; Dave carries an innertube, though...
Moab has obstacles that can get the best of very stout trucks, particularly so when people think they need to be competitive, and testosterone gets in the mix - so even I sounds like anyone's Mom, I'll say it anyways:
I'd say in general it's a good idea to take it down a notch and feel out the place first. There's going to be a bunch of guys who've seen it all, and they won't be impressed anyways, so... The real advantage of the 'Bash is that those same guys are the ones to watch... I'd suggest to start with a trail where the hard obstacles have bypasses, so you can get a feel for it AND have a choice...
Moab also has a bunch of obstacles where there's a fine line between making it and having things go wrong - but if they do go wrong, they can go VERY wrong...in some places, it's a very long way down...
Regarding parts: I'm trying to carry driveline parts and 1/2-ton axleshafts - most of the 1-ton guys carry a u-joint, and maybe a hub, but no spare shafts.
Good battery mounts are important - the most spectacular incident I have seen was one guy bouncing his truck on the wrong line in Pritchett (due to not listening to the spotters!), shaking the battery loose - thanks to the bungee cord battery tie-down! - battery welds itself to the valve cover, and thanks to an ill-functioning carb/PCV and unburned fuel fumes somewhere in the engine, the spark ignites the fuel fumes and blows the valve cover and the air cleaner cover clean off the engine. Some small fire, huge cloud of smoke, everyone running with fire extinguishers, but the enthusiasm to help shrunk considerably when everyone saw the bungee cord...Moral of the story: Murphy's Law rules, everything that can go wrong will - and Moab's gonna find it, and the example above was just asking for it.
I just try to go over everything before I drive to the 'Bash just to make sure it's in good condition. That way I hope I catch the obvious things...
Now this one sounds odd, but Moab trails, while all relatively close to town, are relatively long (for the SheetMetal Gand Golden Spike run, I figure 37 miles and 10+ hours - if nothing breaks). This statement is for those who mostly wheel at offroad parks or your neighborhood 'wheeling spots. If something breaks, you can't just walk back to camp. Parts and tools in camp won't help much out on the trail. Also, fuel is an issue, for those with small fuel cells and thirsty motors, it's a good idea to fill up regularly - my experience is that I use about 1 gallon per hour, so 10 hours on the trail with a 20gal cell means half your fuel is gone. I know this sounds like a no-brainer, but I've been there, and getting asked in the middle of Metal Masher how far it is to town for gas isn't a good thing...