Is it permissible for a Muslim woman to appear in front of a kaafir woman without hijab?
Question

You said that it is permissible for non-Muslim women to see a Muslim woman without hijab. I am not convinced yet, because you quoted one hadeeth which says that ‘Aa’ishah appeared before a Jewish woman. I hope that you can explain the matter further, because my in-laws are not Muslim.
Praise be to Allah.There is nothing wrong with a woman looking at a man; hence men, whether they are Muslim or kaafirs, are not enjoined to cover their faces. Women are enjoined to cover themselves but men are not. Similarly, there is nothing wrong with a woman looking at another woman if one of them is not Muslim. That is because the prohibition on looking is due to the fear of desire being provoked, and it is well known that a woman will not be stirred by desire if she looks at another woman. 
As for the words of Allah (interpretation of the meaning): “It is no sin on them (the Prophet’s wives, if they appear unveiled) before their fathers, or their sons, or their brothers, or their brother’s sons, or the sons of their sisters, or their own women” [al-Ahzaab 33:55], the word nisa’ihinna (their own women) does not apply exclusively to believing women; rather it also includes disbelieving women. If a woman looks at a non-Muslim woman that will not result in evil, so there is nothing wrong with her looking at a non-Muslim woman, especially if she needs to meet with her, such as if she is a teacher or a doctor and so on. It would be too difficult for her to observe hijab before her or to look away from her.