Would a cast iron pot work well for chili? If so, why?
I believe that the "clichéd image" you refer to is popular (and cliché) because of chili con carne's long history of being cooked outdoors in big cauldrons or at least over an open flame. As any camper will tell you, iron is the traditional material of choice when cooking with fire.

As for what actually happens - cast iron is a porous material, unlike stainless steel or aluminum.  The seasoning in cast iron is essentially a layer of baked-in grease that fills those pores and over time absorbs the flavor of everything you cook in it.  It absolutely does impart a characteristic flavour, but what that flavour is depends to no small degree on what's been cooked in it before.  If you don't care properly for your cast iron, that flavour might very well be rancid fat.  So whether or not the flavour you get with your chili makes it better, worse, or just different is largely a matter of personal taste and habits.

As for the acidity: Yes, acids can react with the iron, but cast iron is fairly corrosion resistant.  Cast iron is also used for pipes, so this is pretty well-documented by engineers; you need a pH of 4.3 or less to corrode cast iron to any significant degree.  Pure tomato juice is not a strong acid; it has a pH of anywhere between 4.1 and 4.6, which is just at the threshold; when you consider that this is being diluted with tomato purée, water, meat juices, and various other lower-acid substances, it's evident that you're well within normal tolerances and can cook chili and other tomato products for as long as you like.

Also, the seasoning itself provides some measure of protection for higher-acid foods, since the acid has to penetrate the grease, and water is not good at penetrating fat - that's why food rarely sticks to well-seasoned cast iron.

Now this is not to say that the iron won't react at all, it just won't noticeably corrode or make your sauce/chili taste like metal.  Consensus seems to be that you'll end up with up to 5 mg of iron for every 3 oz of tomato sauce, which comes out to about 160 mg for the whole 3 quarts.  That's not good for you if you eat it all at once, but toxicity doesn't happen until you hit 45 mg or so every day for a prolonged period of time, so unless you're eating an entire quart/litre of chili or tomato sauce a day, it's not going to be a health hazard.

So don't fill your cast iron pot with lemon juice or dump half a cup of vinegar in there - but don't worry about a few tomatoes either.  If anything, it just improves the nutritional content.

As far as heat distribution is concerned, cast iron really does not heat very evenly.  It's far more prone to hot spots than reasonably-priced aluminum- or copper-bonded stainless steel.  What it does do is retain heat very well, so it's excellent at maintaining just the right simmering temperature for something like chili, which you would otherwise need some fairly expensive stainless steel to achieve.  But you have to be very careful of scorching when you cook chili in cast iron; stir often.  A few times I've left my cast iron chili pot sitting around unattended for just a little too long and found some of the meat stuck to the bottom.

Last but not least is the seasoning and the answer is no, you wouldn't season it any differently.  As I alluded to above, the seasoning really doesn't have much an effect when you're cooking a sauce or stew as opposed to a piece of meat or something else that has a tendency to stick.  The only function it really has here is to impart a tiny bit of flavour and insulate the iron from some of the acidity of the tomato juices, which isn't necessary anyway.

All in all, I find cast iron to be very good for chili and other stews primarily because of its heat stability, but it's no panacea, and if you're only using the pot for chili then you won't get many of the benefits (in particular the beautiful "black" seasoning that accumulates after hundreds of rounds of greasing and cooking).  So by all means use your existing cast iron for chili and stews, but I wouldn't recommend for anybody to run out and buy a cast iron pot just for that purpose.