الجواب وبالله التوفيق      

From this institution the Shara’i solution of the issues is advised according to the Hanafi School of Thought and without need it’s not advised otherwise and the other Schools of Thought are not considered at the time of advice.

The issue which has been asked about is not a new issue, the Ummat has been facing this issue since the period of Nabuwwat. The ruling advised about it in the Ahadith at that time is respect worthy and practicable to date the same way. After this necessary preamble now the answers to your questions are written in order:

• It’s not permissible. However, it would be correct to touch with such a cloth or intervening thing which the person touching the Qura’an is neither wearing it nor it is sticking to or sewn to the Qura’an e.g., the Qura’an cover/satchel or any other cloth, etc.

ولا یجوز لہم مس المصحف بالثیاب التی ھم لا بسوھا ۔ عالمگیری کتاب الطہارۃ، فصل رابع

• It’s not permissible for the menstruating lady to read the Qura’an with the intention of recitation, however, if there is a need such as a female Qura’an teacher needs to read for teaching then do not recite the whole verse in one stretch, rather read breaking it word by word.

و إذا حاضت المعلمۃ ینبغی لہا أن تعلم الصبیان کلمۃ کلمۃ و تقطع بین الکلمتین

البحر الرائق ۔ کتاب الطہارۃ ۔ باب الحیض

• As the recitation of the Qura’an is forbidden during menstruation and memorization of the Qura’an is not such a necessity for which one Haram is made Halal, especially when it can be done in the days other than the menstrual period. Therefore, the Hifz will also be forbidden during menstruation. However, if Hifz becomes necessity for someone for any reason then there will be room to read in the way described under the answer for question # 2. Still it would be better to memorize and preserve it by listening to the Qura’an from a cassette or tape.

واللہ اعلم  بالصواب