What is the ruling on the thin, transparent, water-like fluid that is discharged after waking up from sleep, and does it necessitate ghusl every day? What is the ruling on fasting and prayer with it? And what is the ruling on one who took an oath without intending it and without uttering Allah's name?

If a woman is certain that what she sees after waking up is manī (seminal fluid), then ghusl (ritual bath) becomes obligatory upon her. Without this certainty, it is not obligatory.

The discharge might be vaginal secretions, and if she doubts whether it is manī or something else, she has a choice: she can assign it the ruling she wishes.

As for an oath concerning a future matter that was uttered inadvertently without intention, it falls under laghw al-yamīn (vain or thoughtless oath), for which there is no expiation. Nothing is obligatory upon her if she is not certain that the oath was binding.