Hunting Shotgun: Better range, better accuracy, better durability.
Riot Shotgun: High DPS and can effectively shred things at close range, combines with And Stay Back! (Knockdown chance PER PELLET!) and Shotgun Surgeon to make sure the enemies get relentlessly knocked-back into another dimension.
Both of them have the same x1 crit, so it really depends on your build. If you exclusively prefer a sneak range build, the Hunting Shotgun would deliver a more precise payload and better crit in general at the cost of painfully slow reload and generally less awesome... But hey ho, one shot one kill! Otherwise there's no point of using it over the Riot Shotgun.
Second question: I think each pellet does their individual crit (Just like how the http://fallout.wikia.com/wiki/The_Terrible_Shotgun in FO3 works), so when you score a sneak attack all of them will crit; otherwise it's a chancy game with x1 crit chance for each pellet with both the Hunting Shotgun and Riot Shotgun.
Third question: I'm running an all crit perks, max Luck max Agility Guns/Sneak build, and I hardly ever crit with Dinner Bell; while I do most of my work with pistols and Sniper/Hunting rifles. So I wouldn't make shotgun/explosive a crit build, just focus on tanking and getting close. The Riot Shotgun does insane DPS/Knock-back anyway, and with the Big Boomer as back-up, you will be just fine.
BUT consider this: The Tri-Beam laser rifle is a sort of shotgun, which has a 1.5 crit chance. It also spread much less than the normal shotgun, but all three beams have a high chance of criting with a crit build. So it might benefit you to consider a crit build with the Tri-beam as a range option, and switch to the Riot Shotgun for close range, so you can get the best of both worlds... I mean, what else are you going to put your SPECIAL points in anyway? You don't need much Strength for shotguns, and I'm pretty sure that if you're wearing Medium Armor you have the best balance option between mobility and DR; so high Endurance would be trivial.