I'm not going to try to steer you away from a multi-hose hookah, as I understand you want one for camping with friends, but let me just say I have never liked them as much as a good single-hose machine. I will try and help you based on your criteria:
a) Like I said, I have never been a fan of multi-hosers, check valve systems or no. Without check valves, it always seems like someone is trying to take a hit at the same time as someone else, so timing can be an issue. You'd think it would be no big deal, but I have seen it happen countless times. With check valves, the draw is always tighter (I have never smoked out of an auto-seal system that didn't pull a little tight), and I have heard countless stories of auto-seal systems getting gunked up/rusted/just plain stuck for whatever reason. I would go so far as to say that if you must have a multi-hose machine, get one without check-valves, and buy some flip caps like these for people to use when they're not smoking:
b) I would avoid an ice-chamber. Every hookah I've smoked with one on it did not cool down the smoke significantly, if at all. Plus they are messy, are prone to leak, and are just one more thing you have to clean at the end of the day. The concept seems like it should work, but it never has for me, so I dunno.
c) Do not let a carrying case be a make or break item for choosing a hookah. Carrying cases are dirt cheap. Anything that comes with a hookah like you describe is going to be just a cheap, generic, padded duffel-bag type case. Find a hookah you like, and if it comes with a case, great. If not, buy a case like this and call it a day:
d) I think something in that size range would be fine.
That being said, I have always really liked Social Smoke's Egyptians, and they have at least a couple 3-hose setups on their site: