Is it permissible to eat the meat available in shops in Rome, given the doubt as to whether it was ritually slaughtered or died naturally, especially since there is meat described as "halal" but of unsuitable quality and price?

The basic principle concerning slaughtered animals is that they are unlawful until their permissibility is known. If news becomes widespread in a certain country that they do not slaughter according to Islamic rites, then caution dictates refraining from eating or purchasing from them. The criterion for the permissibility of slaughtered meat is the state at the time of slaughter. If the slaughterer is a Muslim or a person of the Book, and adheres to the Islamic regulations, then it is permissible to eat; otherwise, it is not. The way to ascertain this is by asking knowledgeable people and Islamic centers.