Many of these questions are difficult to answer, as they are not explicitly stated anywhere. But here is my attempt nevertheless, based a bit on "Mercenaries Spies and Private Eyes" (pen-and-paper RPG on which Wasteland is based).
Jonesy89: Seems like a lot of skills are dependent upon IQ, might be a problem depending on how the dice roll. When these skills level up, can they make up for a mediocre influencing stat, or is a good starting stat required to get much use out of its related skills?
For the most part, the stat is more important. Skill checks are rolled against the stat, and the skill ranks act as modifiers to the roll. In some cases ranks are very important (such as Brawling, where increased ranks allow increased # of attacks).
Jonesy89: Furthermore, the relationships between some skills, and in some cases the skills themselves, are a little unclear. For example, the wiki mentions that there is a skill for knife fighting as well as a skill for throwing knives; does Knife Fighting overlap with Throwing Knives at all? Does Acrobat allow a character to leap past an enemy in combat or have any other use I should be aware of?
Short answer: don't waste your slots on knife skills :D. Also, I'm pretty sure there is no overlap. I don't think there is any use for Acrobat in combat (it may passively make you slightly harder to hit, but I doubt it), and I think there is only one real use for it in the game outside of combat.
Jonesy89: Finally, do any of the stats affect carrying capacity or what weapons can be used? If I roll up someone who is extremely agile and dexterous, for instance, but with next to no strength, is there any chance of them being able to use a rifle/SMG, or should I designate them as the group pistol/throwing knives expert?
No, none of the stats affect what weapon you may carry or use (but stats do affect your attack rolls and damage - DEX and LUCK for ranged; STR, DEX, and LUCK for melee I think).
Jonesy89: If the leader fails their check, it doesn't move down the list to make subsequent checks?
No, I think it's only based on the leader's skill.
Jonesy89: That said, if a stat is low enough, does it ever start imposing a penalty on the relevant skills it impacts, or does it just mean that I won't get any bonuses on that skill?
As stated above, it actually works the other way around. The check is based on the stat, with the skill being a modifier. In combat, STR, DEX, and LUCK can have an effect, a score of 8 or less gives a penalty, 13 or more gives a bonus.)
