Is it permissible to eat a falafel sandwich that may have been touched by grease from grilled meat, fallen from a shawarma skewer, in restaurants that sell shawarma suspected of not being halal? And is it permissible to buy shawarma when this doubt exists?

The slaughtered animals of thel People of the Book, meaning Jews and Christians, are permissible if slaughtered according to Islamic law. This is based on the Almighty's saying: "This day [all] good foods have been made lawful for you. And the food of those who were given the Scripture is lawful for you and your food is lawful for them." [Al-Ma'idah:5]. Christians are treated as People of the Book even if they are not observant of their religion or are believed to have disbelieved in their religion. However, if slaughtering by non-Shar'i methods (such as stunning to death) becomes widespread and lawful mixes with unlawful, then these slaughtered animals are not permissible unless it is ascertained that they were slaughtered according to Islamic law, because the default ruling for meat is prohibition. As for other foods like falafel, they are permissible, and the possibility of them being contaminated with meat oil is a weak doubt that should be disregarded.