All perfect praise be to Allah, the Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
The ruling on eating the meat of slaughtered animals in the countries of the People of the Book (Christians and Jews), given the prevalence of the methods of slaughtering that do not conform to the Shariah, is subject to a significant difference of opinion among contemporary scholars. Some of them hold that it is allowed as long as the Muslim does not know that these animals were slaughtered in a manner that contradicts the Shariah, acting upon the basic principle in this regard (that all things are lawful unless proven otherwise). Others hold that it is forbidden unless the person knows that they were slaughtered in accordance with the Shariah, acting upon the apparent indication (that these animals are most likely slaughtered in a manner that does not conform with the Shariah).
The European Council for Fatwa and Research (ECFR), chaired by Dr. Yusuf Al-Qaradawi, studied this topic and, after discussions and field research and investigation of the methods of slaughtering used in Western countries, issued the following resolution: "The Council concluded that it is impermissible to consume the meat of poultry and cows; contrary to sheep and young calves, as the method of slaughtering them does not contradict the conditions of Islamic slaughtering in some countries."
Shaykh Ibn ʻUthaymeen was asked, "Is it allowed for a Muslim to consume the meats of the People of the Book in their countries, given that they do not slaughter in accordance with the Shariah?"
He replied:
"Allah, the Exalted, says (what means): {...and the food of those who were given the Scripture is lawful for you} [Quran 5:5] without further details. Hence, whatever they regard as food and which is not unlawful in itself is lawful (for the Muslims), and you do not ask about it, except for what you know with certainty to be unlawful. Pork, for instance: we do not eat it because it is unlawful in itself. Sheep and goats, we do eat them, unless we know that this goat, in particular, was slaughtered in a manner that does not conform to the Shariah. However, some early and later scholars held that even if they (the People of the Book) strangled the animal while believing that it is lawful for consumption, it is lawful for us to eat it because Allah, the Exalted, says: {...and the food of those who were given the Scripture is lawful for you...} [Quran 5:5] So whatever they regard as food is lawful for us, but we do not agree with this opinion. We believe that it is a must to let the blood flow and mention the name of Allah over the animal. However, if a person from the People of the Book offers us meat, we are not obliged – or, rather, we do not have the right – to ask how it was slaughtered and whether the name of Allah was mentioned over it or not. The evidence for this is that ʻAa’ishah, may Allah be pleased with her, narrated that some people said to the Messenger of Allah, sallallaahu ‘alayhi wa sallam, 'There are people who bring us meat, and we do not know whether they mentioned the name of Allah over it or not.' He replied, 'You mention the name of Allah (over it) and eat it.' [Al-Bukhaari] This is a sign from the Prophet that one should not ask. You only need to mention the name of Allah and eat."
As for your other question about the chicken, if we followed the opinion that it is unlawful to eat the chicken slaughtered in Western countries and you know that this meal is most likely mixed with the unlawful chicken, such as if it was cooked with chicken broth or the like, then it is unlawful. If this is not known, then it is not apparent that it is unlawful. It is more prudent and safer for a Muslim, though, to refrain from eating it.
Shaykh Ibn ʻUthaymeen was asked, "Is it permissible for us to eat fish or fries that were fried in oil that was used to fry unlawful meat or, perhaps, pork? Is it necessary to ask the restaurant owner whether he fried something else in that oil or not, or do we just eat without asking?"
He replied, "As long as we know that most of the food cooked in it is impure, such as dead meat or pork, then we have to ask, but if we do not know that, then we do not have to ask."
Allah knows best.