I may be wrong, but I think that the only order that results in the corps's units "rushing" as quick as they can to the destination is when you give a march order.
I think you are correct. I use this often when in the rear and safe from enemy attack, mainly to re-deploy from one area of the battlefield to another.
In all other cases corps go through several phases, namely "concentration", and they march orderly to the destination.
Yes, I believe so as well.
Nevertheless, I am also not sure about the relevance of giving line/column/mixed order orders have when we order a defensive line (the way they march ?? I think not...). But it is a fact that when they form a defensive line the front regiments form in LINE, as Hook stated.
I'm not so sure about this, I will try to check next time, is the first line ALWAYS in line ?. I thought (I could well be wrong) that they deployed in line, column, or mixed order depending which you select.
As to when to use each, I find it depends on if your flanks are safe, the length of line to be covered and what your trying to do.