Your two examples contain two different definitions of "way":
a method, style, or manner of doing something; an optional or alternative form of action.
a particular aspect of something
You should find these two distinct definitions in any good dictionary.
Your examples:
It's amazing the way she manages to stay so calm.
Here, "way" refers to the method or manner of which she stays calm.
In some/many ways it would be better if we met on Monday rather than Wednesday.
Here, "ways" refers to different aspects of the reason why Monday is better than Wednesday.
I'm not fully sure what you mean by using "way" in connection with a choice or opinion, but when you choose to go a particular direction, it can be described as going a particular way - not just literally moving in a particular direction, but also figuratively, following a course of action:
He chose to go one way, I went another.
That is certainly one way of doing things.
In these examples, the first dictionary definition I quoted above applies, so is not really any different to your example.