If there is a legal motion, must there be a legal motion where the moving line passes over each point exactly once?
I think the answer is No.
Consider the following points (co-ordinates written in the parantheses) - A(1, 1), B(0, 0), C(-1, -1) and D(-2, 0). Let them have the following weights -
A: +10 B: -100 C: +500 D: -5
Let's describe the current position of the line by specifying the set of points lying above it. Initially, the line is above all the points. Next, the only option for it is to cross A and reach the position {A}. It cannot cross B now because of its very large negatice weight, so the only option is to cross D and reach the position {A, D}. It can still not cross B. However, if it rotates a bit, gets C above it, pushes A below and reaches the state {C, D}, then it will have a very large positive weight above it and so in the next two steps it will be able to cross B and then A and hence reach below all the points.
Therefore even though there is a legal motion, there is none where the line crosses each point exactly once.
Here's a legal motion for the above set of points. Different positions of the line have been marked with numbers increasing in the chronological order. The initial position is not shown, but you can imagine it to be somewhere above all the points.
