Do excessive movements in prayer—such as taking off a watch, taking out keys, or taking a few steps forward—intended by the worshipper to overcome obsessive whispers, invalidate the prayer?

All the minor actions mentioned in the question do not invalidate the prayer. It is disliked to perform them without need, but permissible when there is a need. Examples of minor actions performed by the Prophet (peace be upon him) and the Companions due to need have been reported, such as carrying children, killing a scorpion, opening a door, adjusting the lower garment (izar), and arranging the upper garment (rida'). These actions do not invalidate the prayer even if performed without need, but they are disliked in such a case.