Hello Marco,
>>>3) At the same time, the resulting curve must contain the EXACT number and position of control points of the original curve.
That is not enough - You should have also the same parameterisation, corresponding knot vector, .....
And that causes the problem: e.g. a radius/circle has a spezial knot vector and parameterisation. The rebuild leads also to a swinging of the curve.
Could you explain were the problem is with the seams?

Have a nice day
Karsten