Do movements during the Tashahhud invalidate my prayer? Does the crossing of worshipers in front of me affect my prayer? Does the Imam bear responsibility for the mistakes of the worshiper in the final Tashahhud?

Disregard these whispers (waswas) and turn away from them. You should recite the Tashahhud normally, and when you finish it and your Imam makes the Taslim, you should make the Taslim immediately, without delay. As for the movement you asked about, it is minor and does not invalidate the prayer. Likewise, people passing in front of you does not invalidate your prayer, though you should hasten to make the Taslim immediately after the Imam. One who suffers from waswas does not perform the prostration of forgetfulness (sujud al-sahw) due to his whispers. So, if waswas occurs to you after your Imam has made the Taslim and you are still in the Tashahhud, do not perform the prostration of forgetfulness; rather, finish your Tashahhud and make the Taslim immediately.