When framing a wall is it really important to have the crown of each stud in the same direction?
Yes it is. If all your studs are oriented the same way, you'll have a gentle bow to the wall (either in or out) that is more or less consistent across the width of a wall.
On the other hand, if you have them so that one stud curves in and the next curves out, you're going to have irregularities in your wall that are the twice the size of the curvature of the studs, and that will be much more noticeable.
Here's an article that explains it well.
Depends on how much the studs crown, but I would say yes. If the studs have a lot of crown to them and they are going in different directions, it can make for a wavy wall and some messy finish work later on.
How much a wavy wall really matters depends on what you are doing with it as well. In a simple drywall wall, you probably wouldn't notice if it was slightly wavy. However, if you put a countertop against it, or try to tile it, even small deviations can be quite noticeable.