What is the ruling on buying meat labeled "halal" from Turkish and Arab stores and restaurants in Germany, which are run by Muslims, given the uncertainty about the method of slaughter or its source?

The questioner can ask the Muslim workers in the restaurant about the source of the meat and the method of its slaughter to reassure himself, and this is preferable. He can also choose not to ask and rely on the presumption of permissibility for what a Muslim sells in terms of meat.

Sheikh Ibn Uthaymeen mentioned three cases for meat imported from the lands of the People of the Book: 1. Knowing that the slaughter was done according to the Islamic method: It is undoubtedly permissible. 2. Knowing that the slaughter was done not according to the Islamic method: It is undoubtedly forbidden. 3. Knowing that slaughter occurred, but being ignorant of its manner (whether the name of Allah was mentioned or not): The default is permissibility, and questioning is not obligatory to ease matters for people and based on the principle of permissibility. This is supported by the action of the Prophet, peace be upon him, who ate from the slaughtered animals of the People of the Book without asking, and by the hadith of A'ishah, may Allah be pleased with her, concerning a people who brought meat and the Companions did not know whether the name of Allah had been mentioned over it or not, whereupon the Prophet, peace be upon him, said: "Mention Allah's name over it yourselves and eat it." And they had recently embraced Islam, which indicates that the default for the actions and dealings of one whose slaughtered animal is permissible is validity and soundness, unless there is proof of invalidity.