Differences in color conversion (RGB to CMYK) / How to choose Corporate Color

I figured out that there are possibly more than one methods to convert RGB to CMYK.

Using Adobe Illustrator CS6 (Mac) the color conversion is completely different as the conversion via Wolfram Alpha. I used the simple cyan (CMYK 1,0,0,0). I already know that most computer displays use RGB to outline the color and I cannot “trust” the CMYK color on my display.

But how to correctly convert a RGB to CMYK color and why displays Wolfram Alpha the color completely different as Illustrator does? Wolfram Alpha shows the cyan very light whereas Illustrator shows it darker. Which one is more “correct”?

I want to have a corporate color and use it on the Web and for some print products. Assuming that my display is calibrated and I like a specific color. Is it better to choose a RGB color and then convert it to CMYK or vice versa?

And how to ensure that the color looks the same everywhere?