Nidá / széfer Torá

Further elaborating on the issue of Torah study while in a state of impurity, it was taught in a baraita that Rabbi Yehuda ben Beteira would say: Matters of Torah do not become ritually impure and therefore one who is impure is permitted to engage in Torah study. He implemented this halakha in practice. The Gemara relates an incident involving a student who was reciting mishnayot and baraitothesitantly before the study hall of Rabbi Yehuda ben Beteira. The student experienced a seminal emission, and when he was asked to recite he did so in a rushed, uneven manner, as he did not want to utter the words of Torah explicitly. Rabbi Yehuda said to him: My son, open your mouth and let your words illuminate, as matters of Torah do not become ritually impure, as it is stated: “Is not my word like fire, says the Lord” (Jeremiah 23:29). Just as fire does not become ritually impure, so too matters of Torah do not become ritually impure.

Any impure person, even [a woman in] a niddah state or a gentile, may hold a Torah scroll and read it. The words of Torah do not contract ritual impurity. This applies when one's hands are not soiled or dirty with mud. [In the latter instance,] one should wash one's hands and then touch the scroll.

All who are impure, even a menstruating woman, are permitted to hold a Torah scroll and to read from it, provided that their hands are not dirty or soiled.

One Who Has Had a Seminal Emission is Permitted In the Recitation of the Sh'ma, containing 1 s'if
All those who are ritually impure may read from the Torah, recite the Sh'ma, and pray [the Amidah] - except for the baal keri (one who has an emission) who was excluded by Ezra from the rest of the ritually impure, and he [Ezra] prohibited him (the baal keri) from words of the Torah, the Recitation of the Sh'ma, or prayer [i.e. the Amidah] until he immersed, so that scholars would not be found to be [frequently intimate] with their wives like roosters. And afterwards they annulled that enactment, and they stood the matter on it's [original] law, that even the baal keri is permitted with regard to words of Torah, the Recitation of the Sh'ma, and prayer [i.e the Amidah] without immersion and without washing with nine kabs [of water], and such is the widespread custom. Rem"a: There are those who wrote that a menstruant woman in the days when she sees [blood] may not enter the synagogue or pray [the Amidah], or mention the Name, or touch the [Torah] scroll (Hagahot Maimoni ch. 4). And there are those who say that she is permitted in all [of those], and that is the correct opinion (Rashi, Laws of Niddah). But the practice in these lands [Ashkenaz] is according to the first opinion. And in the "clean days" [after menstruation, but before immersion] it is permitted [to do these things]. And even in a place where they are stringent during the High Holidays and similar days, where many assemble to go the synagogue, they are permitted to go to the synagogue like the rest of the women, for it would cause great sadness for them, that all are assembled and they stand outside (Terumat HaDeshen, Part II 132).

In our countries the custom is that it is permitted (for women) to say blessings and pray, nevertheless they shouldn't look at the torah scroll when they are lifting it up (Hagbah) for the congregation to see. It is also written that they (women) shouldn't go into a cemetery until they immerse (In a mikvah)

During the time a woman is a niddah, during her menstruation period before the white days, it is customary for her not to enter a synagogue and not to pray. But on the Days of Awe, that is from the first day of Selichot when multitudes assemble at the synagogue, and she will be greatly distressed if she does not go, she is permitted to go and pray. Similarly at the marriage of her son, or daughter, or she has given birth and it is time for her to go to the synagogue, or for similar things, she is permitted to do so.