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.
Excuse me, but would you update the game, please? The devs wrote the update info on the description, instead of the dedicated update log. 敗北の追憶 ヴルツェルシスターズ to ver1.0.1