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.
There is no doubt that in principle a woman should stay in her home as per the verse mentioned in the question so that she would not cause temptation to men nor be tempted by them by her going out.
Nonetheless, it is permissible for a woman to go out for her need as per the authentic Hadeeth in which the Prophet said: “Indeed, Allaah has given you (O women) permission to go out for your needs.”
Moreover, seeking knowledge is one of the most virtuous acts about which there are many Sharee’ah texts (i.e. from the Quran and the Sunnah). This virtue is not exclusive to men; rather, it is for all, for men and women. Therefore, it is permissible for a Daa’iyah (caller to Islam) to call women and urge them to attend circles of knowledge. For more information, please refer to Fataawa 85903 and 84067.
However, it is not permissible for a woman to go out without fulfilling the Islamic requirements, like seeking the permission of her husband, wearing Hijab and so forth. For more benefit, please refer to Fataawa 82359 and 82401.
As regards the prayer, it is better for a woman to pray at her home as the Prophet said: “…And their (women’s) homes are better for them.” [Al-Bukhari]
Therefore, it is not preferable for a Daa’iyah to call women to the mosque only for the prayer. For more benefit, please refer to Fataawa 136492 and 92531.
Besides, it is not from the guidance of the Prophet nor the Companions among whom are the four Caliphs, to set a special mosque for women to pray in it separately from men.
As regards women gathering in a place limited to them, during Ramadan or at any other time, for this period [that you mentioned in the question] or any other period, for a valid purpose, like for Iftaar (breaking the fast) and the like, then it is permissible for them to do so as long as this does not lead to committing a prohibition or neglecting an obligation. Indeed, such gatherings may lead to many advantages and benefits, like amity and love between them and the like. For more benefit, please refer to Fatwa 153615.
Allaah Knows best.