First a brief explanation...

A picture style/profile is simply a recipe; it is a means of interpreting the raw data of the sensor. It dictates the tone, contrast, sharpness, brightness, and other ways of interpreting the raw data. When shooting jpeg then these recipes are used in the conversion of the raw to jpeg in camera.

So to answer your questions...

1. How does it work? Like a recipe. For programs that read the recipe totally (View NX/Capture NX in the case of Nikon) the raw data will be presented with a particular view. This view will look identical to what you would have seen if the camera captured files in jpeg instead of raw (the raw processing of the recipes is the same in camera as in the NX programs). The difference is that if you capture in a raw format you can change your recipe. If you capture in jpeg you can't change your recipe but start altering pixels right away. Other programs, like Adobe Camera Raw/Lightroom have profiles that mimic the camera maker's profiles but they are simply recipes that work on raw files as well.

2. The only difference it will make is if you use a manufacturer's products versus an alternative. Nikon's NX products read the built in recipe and start there but you can change it to whatever you want (for raw). Other products will not start with the same image but you can usually recreate it if you want to -- or go in a completely different direction. Each program has the potential to display the raw data differently so using different programs can give you different looks for the same raw file. I would recommend only using programs that interpret the raw data identically so you can consistently evaluate your pictures.

3. Some people do shoot only with vivid. But the high contrast curve isn't necessarily suitable to all subjects. Also the higher saturation isn't usually flattering in portraits. It is similar to shooting with Velvia. But that choice, like others, is in the photographer's vision. But no reason not to use it as a standard picture setting for raw if you like it. Some people use the Landscape profile because it has the saturation of vivid and the contrast of standard.

4. My understanding is that there is no hidden data. Simply encrypted settings. And those are usually available via a manufacturer-provided SDK. The thing is that other raw converters, for varying reasons, opt not to use the manufacturer SDK in interpreting the raw data.