1) What period of the Islaamic calendar that Muslims are not allowed to fast?
2) Is it permissible for is Muslim woman to sleep with her husband before purifying herself on completion on her period?

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger. We ask Allaah to exalt his mention as well as that of his family and all his companions.

It is permissible to fast any day of the month all year long. On the other hand, it is not permissible to fast on 'Eed Al-Fitr and on 'Eed Al-Adhaa, and during the three days of Tashreeq: 11, 12, 13 of Thul-Hijjah, unless one is a pilgrim who cannot offer sacrifice, as it is permissible for him to fast on these three days. It is also forbidden for a woman to fast while she is menstruating or during her postpartum time, or voluntarily fasting without the consent of her husband. Moreover, it is disliked to fast a Friday alone unless it corresponds with a day that someone usually fasts, like the person who fasts every other day. It is also dislikeable to fast Saturdays only.

As regards your second question, it is not permissible for a husband to have sexual relations with his wife after she becomes pure from menses and postpartum unless she takes a ritual bath (Ghusl) as Allaah says (interpretation of the meaning): {And they ask you about menstruation. Say, “It is harm, so keep away from wives during menstruation. And do not approach them until they are pure. And when they have purified themselves, then come to them from where Allaah has ordained for you. Indeed, Allaah loves those who are constantly repentant and loves those who purify themselves.”}[2:222]. The saying of Allaah {until they are purified from menses} means after they have taken a ritual bath.

Allaah knows best.