What is the ruling on one who neglected to ascertain the time of the onset of purity from menstruation and to perform ghusl immediately, or prayed while obliged to leave her home and with her feet uncovered, or prayed with a purity that was doubtful despite hastening to perform it, or had natural discharges and prayed in haste? And is it obligatory to make up the prayers in these cases, or should she increase in supererogatory prayers and repent?

Here is a summary of the answers to your four questions:

1. A menstruating woman monitoring her purity (tahara): A menstruating woman must check for her purity upon going to sleep, in the morning, and before every prayer. If she neglects to do so, and then discovers her purity after the prayer time has passed, she must perform ghusl (definition) (ritual bath) immediately and make up the missed prayers, along with repentance.

2. A woman praying with uncovered feet: A woman's feet are considered awrah (parts that must be covered) according to the majority of scholars. If she prays with them uncovered, she is required to repeat the prayer as long as its time remains. However, if she cannot find anything to cover them with and fears the prayer time will expire, she is not required to repeat it, though it is recommended to repeat it if possible.

3. Praying with doubt about purity: Prayer is not valid if there is doubt about the fundamental state of purity; rather, purity must first be ascertained. If something exits the body during prayer (such as secretions), the wudu (ablution) is nullified, and the prayer becomes invalid.

4. Rushing in prayer: It is not permissible for a Muslim to rush in prayer in a way that violates its pillars, such as طمأنينة (tranquility/composure). If one does so, their prayer is invalid and must be made up. As for rushing that does not violate the pillars, it is permissible, especially if there is a need for it.