Why, when you are doing no stirrup work, my instructors tell me to cross the right stirrup over the withers first, and then the neck?

Safety.
Your left over loop of rein should hang on the right[off] side of the horse. When you cross your stirrups in front of the saddle,pull the buckle down a few inches from the bar and cross right over then same for left over. If you fall your rein can only go under the left iron which doesn't have anything over the top of it adding weight and should come free easier.
I am unclear as to the second part of your question.
To ride without stirrups properly, they should be completely detached from the saddle. Otherwise you do get a slight push on your thigh from them, an inconvience not a true hindurence.
They should rest flat and as close to the front of the saddle as possible, not pushed forward up the neck.