Also, it's not as simple as that. You might think "I'm eating less so my body must burn fat" but your body is thinking "shit, I'm starving, better conserve energy" and turns down your metabolism, prioritizes fat storage, etc. A whole bunch of survival mechanisms kick in. Cutting more than 500 calories/day doesn't lead to sustainable results. There's no quick fix.
Sure it does, if you KEEP IT low (enough). That's the problem with viewing a diet as something you do for a while, and the return to what you've always eaten. If you want to follow a diet to get permanent results, you need to do it for the rest of your life.
Yes, you need to keep to A diet for the rest of your life to avoid adding weight back, but if you cut reasonably low amounts of calories of your maintenance level you will be able to go back to a much higher level when you've reached your target weight than if you cut a high amount of calories to get your weight loss.
The currently most hyped diet in Sweden (and possibly other parts of the world too?) is LCHF - Low Carb High Fat (a variation on the Atkins and GI theme if I've understood it correctly) claims to be different from other diets.
From what I've understood the idea is to eat practically no carbs, which will trigger something called ketosis in the body, which in turn supposed to be good.
Read all about it at http://en.wikipedia.org/wiki/Low-carbohydrate_diet
Whatever you do, don't start a diet debate with LCHF fans, they tend to be fairly fanatic :) (although in a curteous way, they'll overwhelm you with references to various studies, papers and whatnot).
Since when did "vitamins" replace "fruit and vegetables"?
Fruits and vegetables has calories too. Look at tables: http://www.fatfreekitchen.com/calories.html
1 apple is about 200g (five apples per 1kg), thus it is equivalent to 2 tablespoons of sugar.