Question Answer

What is obligatory upon a Muslim with respect to non-Muslims concerning different types of interactions and also with respect to holidays and festivals?

The responsibility of a Muslim towards non-Muslims are many, including:
First, he must call them to the way of Allah. This is to preach to them and to make clear to them the reality of Islam, according to his ability and if he has the knowledge to do so. This is the greatest and best good deed that one could do toward his fellow citizen and for those who live together with Jews, Christians and other disbelievers. On this point, the Prophet (peace be upon him) said, "The one who guides to good gets the same reward as the one who performs it." [Muslim] The Prophet (peace be upon him) also told Ali, when he was sending him to Khaibar to encounter the Jews, to invite the Jews to Islam. He told him, "By Allah, if Allah guides one person by you, it is better for you than the best types of camels." [Bukhari, Muslim] The Prophet (peace be upon him) also said, "Whoever calls to guidance will have a reward similar to the reward of the one who follows him, without the reward of either of them being lessened at all." [Muslim, Ahmad, Aboo Daawood, an-Nasaa'ee, at-Tirmidhee, Ibn Maajah] Calling them to Islam, preaching Islam to them and advising them concerning that is one of the most important deeds and is one of the best ways to get closer to Allah.
Second, [the Muslim] may not wrong the other person with respect to his life, wealth or honor, if the non-Muslim is a citizen of the Islamic state or has attained other protection. He must fulfill the other's rights. He may not wrong him with respect to his wealth by stealing from him, deceiving him or cheating him. He cannot harm him in his body by beating or killing him. His protection from the state guarantees his safety from such things.
Third, there is no prohibition concerning buying, selling, renting or other such business transactions with them. It has been authentically reported that the Prophet (peace be upon him) bought things from the polytheistic disbelievers. He also purchased items from Jews. In fact, when the Prophet (peace be upon him) died, his shield was being held as collateral with a Jewish person in exchange for food for his family.
Fourth, one should not give them the greetings of peace first. However, one responds to their greetings. The Prophet (peace be upon him) said, "Do not give the greetings of peace first to the Jew or Christian." [Muslim] He also said, "If the people of the book give you the greeting of peace, then say, 'Wa Alaikum (and upon you)."' [Bukhari] Therefore, a Muslim does not greet a non-Muslim first. But when a Jew, Christian or other disbeliever greets him, he replies, "And upon you also," as the Prophet (peace be upon him) has ordered. This is from the rights of conduct between a Muslim and a disbeliever. He also must be neighborly toward his non-Muslim neighbor. If your neighbor is good to you, you do not harm him and you may even give him charity if he is poor or give him a gift if he is rich. You may also advise him concerning what is good for him. All of this may lead him to want to learn about Islam and become a Muslim and because neighbors have very great rights. The Prophet (peace be upon him) said, "The Angel Jibreel (Gabriel) kept advising me concerning the neighbor until I thought he was going to inherit [from his neighbor]." [Bukhari, Muslim] Allah also says in the Quran, "Allah does not forbid you to deal justly and kindly with those who fought not against you on account of religion and drove you not from your homes. Verily, Allah loves those who deal with equity" [al-Mumtahinah (60):7].
It is recorded in an authentic hadith from Asma bint Abu Bakr that her mother, who was a polytheist, visited her and asked for her assistance - this was during the time of the peace treaty between the Prophet (peace be upon him) and the disbelievers so she went to the Prophet (peace be upon him) to ask him about that and he told her to keep the ties of kinship with her and be righteous towards her.
Fifth, a Muslim should not participate with them in their greetings and festivals. However, one may give them condolences upon the death of someone if he finds some legal benefit in doing so. He may say to them, "May future bring good to you," or something of that nature. He may not say, "May Allah forgive him," or, "May Allah have mercy on him," if the person who died was a disbeliever (see Quran 9:113). That is, one may not make prayers for a dead disbeliever although one may ask for guidance for those who are alive and so forth.

Are women allowed to hold their Jama'ah in normal days if they so desire? Can women hold their separate Jama'ah on Eid days or on Fridays?

It is not advisable for women to hold their jama'ah. The Holy Prophet Sall-Allahu alayhi wa sallam, has always preferred it for women to offer their prayers in their homes. Some Muslim jurists have held the jama'ah of women as a makrooh practice. However, if some women overlook this principle and hold their own jama'ah the female Imam should stand in their midst, and not in front of other 'muqtadi' women, as a male imam would do.
However, in the case of jumu'ah and Eid the jama'ah of women is neither permissible nor valid. (See Radd-al-Muhtar v.l. p.600) It means that if some women held their jama'ah (without men) it will be void and they will have to perform Zuhr prayer again.

Is it allowed for women to do Itikaf at home for the last ten days of ramadhan? What are the rules of itikaf and is she allowed to spend time playing with her grandchildren and socializing with her family?

Yes, it is permitted for women to do i`tikaf (spiritual retreat) in a specific place in their house, whenever they have the time and are able to do so without neglecting their family duties and other responsibilities.
The same considerations would apply as in any other i`tikaf: one's primary concern during the time should be towards worship, and unnecessary worldly activity should be avoided.
Even when one is in their monthly period, it is recommended in the Hanafi school to sit in a designated place of prayer (musalla) at home, after having made wudu, and make dhikr for the time it takes to pray.
It is recommended for women to designate a place in their houses as their 'masjid'. They can do i`tikaf there at any time, even for a brief period of time, [Radd al-Muhtar] and attain the great rewards mentioned in the hadiths for i`tikaf (as long as they fulfill their other worldly and religious duties, as is the case for men).
The Prophet (Sallallahu Alaihi Wasallam) said, "Whoever stands the nights of Ramadan in prayer out of faith and seeking reward shall have their previous sins forgiven." [Bukhari & Muslim]
May Allah give us success to follow the guidance of His Beloved (Sallallahu Alaihi Wasallam).