How to season iron cookware on gas stove?

I strongly advise against doing it. I tried stovetop seasoning at home and got terrible results.

A stove gives you hot spots - on gas, this will be the ring where the flame touches the metal. The temperature of the metal in this hot spot is way too high, and the oil burns instead of polymerizing. You get some oil-charcoal in this place, which doesn't have non-stick properties, and flakes off after a few uses.

Outside of the hot spots, the temperature is not high enough. The oil doesn't polymerize thoroughly, and forms a sticky paste instead of a smooth one. Your food will stick to these parts of the pan even worse than to the charcoaled parts.

Conclusion: use an oven. If your oven is too small for a pan (I only have a toaster oven, 30x30 cm, and my pan+handle is way too long even for the diagonal), leave the door cracked and seal the crack with alu foil. This is not very energy efficient, but you only do it once per pan. It worked for me, and I got a real, non-stick seasoning after multiple failures on stovetop.

I agree with previous answers that the primary seasoning should be done in an oven. However, to respond to the updated question:

The seventh and eighth point in the above link talk about repeated seasoning on the gas stove after every use. How viable is that?

I do that a lot. I don't do it after every use, particularly if I've just used the pan for frying or something else involving oil. But if I've had to really scrape the pan to clean it or have used ingredients (e.g., acids) that don't do well with a cast iron finish, I find the technique mentioned to be very helpful in maintaining a smooth non-stick surface. Just put a tiny amount of shortening (better I think) or oil in the pan or on a paper towel, heat the pan, and wipe it around. Wait until it starts smoking, and remove from heat. Be sure to wipe out excess oil with the paper towel. (If you're careful, you can wad up the towel and do this with your hands; if not, use tongs to hold it.)

By doing this technique repeatedly over a couple months, I even was able to restore a pan that had been deeply scraped and scratched with a metal utensil (by a friend who didn't know better) which disrupted the even, non-stick surface.

I've also found that the stovetop technique gradually helps to smooth out the interior pan surface and improve its quality over time. (This is particularly true of newer cast iron pieces that aren't machined smooth like older ones were and have interior pitted surfaces.) Yes, you could do it through repeated baking and seasoning in the oven, but if you're patient, in a year or so of babying your pan, you'll end up with a surface that seems more durable and smoother than I've obtained through other methods.

All of that said, be careful just to use a small amount of shortening/oil each time and don't use high heat or you'll encounter the problems mentioned in other answers.

I have a number of cast iron pots and pans and have never seasoned any of them in the oven. To do it on the stovetop requires low heat for a longer time. The metal will get hot eventually. If you use a high heat, you'll get hotspots where the oil burns, which is not good.

My best-seasoned pan is the one reserved for eggs, which sees only butter rather than oil, and has a spectacularly good finish. It was a simple matter of frying one or two eggs in butter in that pan every weekday for 5 years :-). Just to say there is more than one way to season a pan.