Mmm, yeah. It doesn't have to be jelly-like just so long as it isn't browned too much on the outside, it's fine. The Japanese add sugar to their omelette too since they want it slightly sweet. Sake/mirin too. I cook mine enough so that it doesn't have any liquid oozing out.
You'd want to use a non-stick pan. Teflon pans aren't exactly non-stick so you'll have to check. But the right amount of oil/butter should do the job just fine, Nanashi.
Yeah. I make mine based off the omelette for omurice which is also how the French omelette is also. The British omelette is different though. It's flat like a pancake but still has consistency and texture.
As Jacques Pepin says, one is not better than the other. It just uses a different method to make.