Ha, sorry, I assume people familiar with the jargon around here.
On the first question, yes they are really design choices. With optical design, as with many engineering problems, there's a balance of trade-offs. It's impossible to 100% eliminate distortion, so different manufacturers make different choices. Same is true for color balance, different brands optimize for slightly different "looks" that are relatively consistent among different models.
As a very sweeping generalization:
- modern Swarovski tends to be the most neutral and bright due to flat, high transmission across most of the visible spectrum, but with slightly reduced contrast and saturation (due to a falloff in the deep red wavelengths);
- Leica and Nikon EDG tend to be a bit warmer and more saturated and contrasty (due to extended transmission well into the deep red);
- Zeiss tends to be a bit cooler/greener, which aids subjective brightness and low light performance as they optimize for the wavelengths in the center of the visible spectrum where our eyes are most sensitive.
On the last question, as tenex noted AMD = Angular Magnification Distortion; this article covers it well (you can skip the math equations if that's scary, the description and images are very clear):
http://www.holgermerlitz.de/globe/distortion.html
Because Swarovski aggressively corrects curvature so that lines appear straight across the field, it creates a phenomenon colloquially termed "rolling ball" when panning where our eyes/brain perceive space as curving over a round surface. At the opposite end of the spectrum are binoculars with a ton of pincushion, when it gets really excessive you can get an opposite effect "rolling bowl"
People vary in how sensitive they are to these effects, but IMO the Swarovski is too aggressive, and the Nikon EDG strikes a better balance between "flat field" and having it look unnatural. Others differ, that's why we have choices!