1- yes, good shoes are important when running. you can avoid impact related injuries with good shoes among other things.

2- i've had nike, reebok and adidas running shoes, all work as intended, each brand has its on price range and usually the $600 while not really superior to the $500 pair is miles ahead of the $200-300 pair. i personaly like my nike's the most since i have wide feet and they fit me better than the others.

3- you look for running shoes if you're going for jogging or running.

4- the Nike Air really does help with impact absorbtion, not necessary but definitely a + since you say you're obese it will help you save your knees and ankles among other things. other brands have some sort of air system aswell.

5- well, the left side of your tummy getting sore is normal, you're just having side stitch, you can find more info at http://en.wikipedia.org/wiki/Side_stitch .
your burning feet imo is normal, not sure what causes it, someone probably does and will chime in but i get burning feet after 5-6miles so... yeah.

also you said that you would be jogging in a stadium?
most tracks in stadiums are made of astro turf or some other type of soft material to provide cushioning, you might want to look into shoes which are designed specifically for use on a running track.

another thing to remember is that you have to replace your shoes fairly often if you use them alot, every 200-450 miles or so depending on the condition of the shoe. so before buying your shoes know how much you're going to be running and budget out your purchases so you dont go with an expensive $600 then wind up having to replace them 14 months later with a crappy $150 pair.