What is the ruling on dealing with the name of Allah written on the ground or in newspapers that contain Qur'anic verses or the name of Allah? Is it obligatory to hasten to remove them, and is it sufficient for one person to remove them to lift the sin from everyone?

Everything that contains the mention of Allah must be respected and protected, and not merely refraining from lifting what contains the mention of Allah from the ground or a shelf out of apostasy, unless there is an intent to disrespect it. It is sufficient for you to advise against disrespecting what contains the mention of Allah, and you are not obliged to pick up every scattered piece unless it is easy to do so without hardship. This act is fard kifaya (a collective obligation). Your question indicates waswasa (obsessive whispering), so we advise you to supplicate, distract yourself from it, and consult a psychiatrist.