Question: Can a husband and wife have Intercourse during the month of Ramadan?
Answer: Yes, it is perfectly okay and permitted for a husband and wife to have sexual intercourse during the blessed month of Ramadhan after sunset (iftar) and before true dawn (al-Fajr al-Sadiq), meaning when they are not fasting.
Allah Most High says:
“Lawful for you, on the night of the fasts, is the approach (sexual intercourse) to your wives. They are your garments and you are their garments. Allah knows what you used to do secretly among yourselves; but He turned to you and forgave you; so now cohabit with them (your wives) and seek what Allah has ordained for you, and eat and drink, until the white thread of dawn appear to you distinct from its black thread. Then complete your fast till the night appears; but do not cohabit with your wives while you are in retreat in the mosques. Those are Limits (set by) Allah. Approach not near thereto. Thus does Allah make clear His Signs to men: that they may learn self-restraint.” (Surah al-Baqarah, V: 187)
In the above verse, Allah Most High clearly mentions the permissibility of sexual intercourse with one’s spouse during the nights of Ramadhan. In the beginning of Islam, the ruling was that if one went to sleep or offered his Eisha prayer, one could not eat, drink or have sex with one’s spouse until sunset the following day. This proved to be somewhat difficult for the Muslims, hence thereafter Allah, the Almighty and Wise, revealed the abovementioned verse, in that it is permissible for one to eat, drink or have sex at night until true dawn comes in, after which one’s fast commences. Some Companions abstained from sexual intercourse during the whole month of Ramadhan, hence the abovementioned verse was revealed. (See: Tafsir Ibn Kathir, 1/298)
However, it should be noted that in the last part of the above verse, Allah Most High prohibits sexual intercourse whilst one is in I’tikaf, hence it is unlawful for the one who is in the state of I’tikaf to cohabit with his spouse whether in the Masjid or outside the Masjid.
As far as your second question is concerned, it is perfectly permitted and lawful for a legally married couple to have sex on Eid day, as there is no fast on this day.
And Allah knows best