Assalaamu alaykum.
1. I would like to know if there is a tolerable/permitted quantity or percentage of haram meat or meat extracts in food. There is a tolerance of alcohol up to a "rate of 5 in ten thousand" according to http://www.irfi.org, so does this go for meat too?
2. If the packaging used for a halal food product contains extracts of animal fat of unknown origin, does that also make the halal food product haram?
3. What would be required to make a food product halal if the packaging is the only thing that is haram?

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, sallallahu ‘alayhi wa sallam, is His slave and Messenger.

We did not find any proof on which we could rely in estimating the exact percentage of tolerable content that you ask about. The relevant statement that we can mention here is that if an insignificant amount of impurity is mixed with a large quantity of a pure substance such that the impurity is totally absorbed and does not affect any of the characteristics of the pure substance in terms of color, taste, or smell, then the resulting substance is not declared impure. The same ruling applies to the case when the impurity is transformed into a totally different substance.

Shaykhul-Islam Ibn Taymiyyah wrote:

"Allah, the Exalted, deemed the evil things such as blood, dead animals, pork, and the like unlawful. If these things fell into water or any other liquid and dissolved completely such that no traces of them can be detected, then the water or liquid is declared pure. The same applies to alcohol when it dissolves entirely in a liquid (and no traces can be detected in terms of taste, color, or smell), then the person who drinks this liquid is not drinking alcohol. If the alcohol turned into vinegar naturally, it is declared ritually pure according to the consensus of the scholars. This is a stronger view with regard to those who argue that when the impurity is transformed into a different substance it becomes pure, such as Abu Haneefah and Ahl Ath-Thaahir (those who interpret the texts literally and refuse to accept clear analogy), and it is one of the two opinions of Maalik and Ahmad. The turning of the impurity into salt or ash and the like is the same as its turning into water (as far as the religious ruling is concerned); there is no difference if the impurity turns into ash or salt or dust or air or the like. Allah, the Exalted, has deemed the good and pure things lawful; these kinds of grease and milks and sweet and sour beverages fall into this category, and the impurity has transformed; how then could we deem unlawful what Allah has deemed lawful! There is no evidence to support the claim that if the impurity is dissolved in the pure substance, it is declared impure and is unlawful. There is no evidence of that, neither in the Quran, nor in the Sunnah, nor in the consensus of the scholars, nor in analogy."

Ibn ʻUthaymeen said, "As for the percentage, you should not think that any percentage of alcohol in a thing makes it unlawful; rather, if the percentage of alcohol has an effect whereby when a person drinks this mix he becomes intoxicated, then it is unlawful. But if the percentage is very small and without effect, then it is lawful. For example, a percentage such as 1%, 2%, or 3% does not make something unlawful..." [Liqaa' Al-Baab Al-Maftooh]

Determining whether the impure substance has dissolved and was completely transformed is decided by the specialized experts. Mere doubts that a given substance may be unlawful or ritually impure are not sufficient to deem its consumption and usage unlawful, and food must not be thrown away based on mere doubts.

Ad-Dardeer wrote:

"If an impurity fell into a solid substance, such as solidified ghee, Thareed (a dish made of meat and bread), or solidified honey, or if a mouse died in it, then it is deemed impure only if one thinks that the impurity has spread in the whole quantity because it remained in the food for a long time; otherwise, if one thinks that the impurity has not spread in the whole quantity, then only the part which one thinks the impurity has spread to is deemed impure. This differs according to the fluidity of the impurity and the amount of time it remained in the pure substance. Only the parts one suspects has become impure should be removed and thrown away, and the rest is pure and lawful for use – even if one suspects that the impurity has spread throughout the pure substance – because food must not be thrown away based on mere doubts." [Ash-Sharh As-Sagheer]

From the above, we deduce the ruling on the packaging that is used for lawful food products and contains animal fat of an unknown origin. The basic principle is that all things are lawful. Moreover, this packaging is often subject to several factors that cause its transformation, and according to the most likely correct view, as mentioned above, transformation purifies it.

Ibn Taymiyyah said, "The view that it (the impurity) becomes pure by transformation is more correct, because if the impurity becomes salt or ashes, then its essence, characteristics, and name have changed. The religious texts prohibiting dead animals, blood, and pork do not refer to salt or ash or dust, neither in clear terms nor in meaning. The reason for which those things were ruled as unlawful is non-existent in these ones, so there is no basis for saying that they are unlawful and impure."

Ibn Al-Qayyim wrote:

"Allah, the Exalted, causes the evil to emerge from the good and the good to emerge from the evil; the origin is of no effect in this regard; rather, it is the nature of the new substance that determines whether it is good or evil. It is inconceivable that the impurity remains despite the fact that its name and description cease to exist (when the impure substance is transformed into a different substance). The religious ruling is subsequent to the name, and the description is connected to the name (if the substance is transformed into another substance, its description changes and thus the ruling on it changes)." [Iʻlaam Al-Muwaqiʻeen]

Allah knows best.