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.
Cast iron care need not seem so intimidating or mystical. There are lots of ways to take care of it, and though many will profess their own gospel and taboos associated with cast iron cookware, many different solutions will work. You just need to understand some of the basic principles and the rest is personal preference.
First, in my naivete I used vegatable oil as the seasoning
There is nothing wrong with using vegetable oil as your seasoning. That is often what you will be cooking with anyway, and every time you cook with oil you are contributing to the seasoning on your skillet - so obviously it is a fine type of oil to constitute your seasoning.
second, I also didn't remove the excess oil, leaving a sticky and uneven finish.
This has happened to me a number of times, even after removing excess oil. When you season it in an oven (preferably upside down), the oil heats up which can often cause it to run more easily before it has a chance to bond to the iron. That can cause it to leave spots without seasoning, and the spots that are seasoned will sometimes be sticky. I would just repeat the process again until the surface is completely seasoned. I usually wipe a light layer of vegetable oil on mine, turn it upside down in the oven, and bake it for an hour or two at around 350°F with a cookie sheet under it. It'll probably smoke a little bit, but don't worry about it.
Additionally, when I break in a new cast iron whatever, I like to cook with really greasy and oily foods (like bacon) the first few times I use it. It will help create a better seasoning and get rid of those sticky spots. Really let it pool up and try to cook for a decent amount of time. That can help get a nice, uniform finish on it. Scrape it clean with a lightly abrasive sponge or nylon brush and rinse it in cold water (maybe a tiny amount of dish soap if you really want, but it honestly doesn't need it). Dry off. Cook on it again. The more you cook with it, the better the seasoning will get.
Looking at your pictures, I would not do anything so drastic as use a caustic cleaning chemical to strip the seasoning off and start over. Just scrub it nice and clean with some soap, dry it off, and try seasoning it again. And again if you are still not satisfied. You'll get there.
Best Answer
The PH scale ranges from 0 - 14 with the lower numbers being more acidic than the higher number. Canned tomatoes are generally around 3.5-4.5 on the acidic scale, in other words just above the half way point below neutral 7.
As long as your tomato based sauce is not left in your seasoned Kadai (as Kristina rightly points out) for a long timeframe and your kadhai is washed out immediately after use no damage to the seasoned coating should occur.
Just remember to wash out your Kadai (with warm water and a cloth) immediately after use to preserve the seasoned coating.