Does one become an infidel (kafir) if he sees a wall on which "Bismillahi ar-Rahmani ar-Rahim" is written, and the lower part of it is stained with car oil or filth, and he does not change it? And is he obliged to buy paint and change it without the owner's permission or if he does not know the owner? And does one become an infidel if he sees a paper on which an ayah (verse) is written, thrown on the ground, and he does not pick it up? And does the ruling differ between the name of Allah, an ayah, and a Mus'haf (copy of the Quran)?

What you mentioned is neither an insult to the Basmalah nor an act of disbelief (kufr). These elaborate distinctions stem from obsessive doubts (waswasah), and it is advisable to disregard them. There is no difference between something that contains a verse or a name of Allah, according to those who deem it kufr to not lift what is found in dirt. So, throwing a Mushaf into dirt or leaving it there is considered kufr by the Malikis. This includes the names of Allah, the prophets, and hadith. However, some scholars believe that such an omission is not kufr but merely a forbidden act (haram).