After trying to recalibrate one point, why do all the other points change?
The lines emanating from the calibration points are an indication of the calibrated accuracy for the associated region of the screen (shorter lines are better). This is not necessarily an indication of which calibration points should be improved; as you have seen, improving a single point can change the accuracy in other regions of the screen.
The accuracy results show the user which areas of the screen will be more accurate, not necessarily the points where the user needs to improve their calibration.
Ending up with several bad calibration points is common when working with users that have a hard time maintaining gaze fixations for the requisite amount of time during calibration. For these users it is helpful for the facilitator to watch the user's eyes during calibration and make a note of the points for which the user had a hard time maintaining fixation. Those points can then be improved on a second pass, rather than relying on the accuracy display to select points for improvement. Watching the user's eyes during a calibration rather than watching the screen is the best technique when a user is calibrating.