Why was the exact date of Laylat al-Qadr concealed?

The wisdom behind this is to encourage Muslims to strive in worship throughout all the nights of Ramadan, or at least in the last ten nights. By doing so, they will earn the reward of observing Laylat al-Qadr as well as the reward of worshiping on other blessed nights.

What is the ruling on the fasting of a woman who has reached menopause if menstrual blood flows?

If a woman reaches the age of menopause (which is usually sixty-two) and her menses have ceased, then she sees blood after that, and its duration is not less than a day and a night (24 hours), it is menstruation (hayd). If it is less than a day and a night, she is considered as having non-menstrual vaginal bleeding (mustahada), so she fasts and prays. However, she must perform ablution for every obligatory prayer after its time enters, pray immediately, and be treated as a person with a continuous condition. There is no specific end limit for a woman's menstruation; it is possible as long as the woman is alive. And Allah the Almighty knows best.

Does the time of Fajr (dawn) prayer start after the first Athaan (call for prayer), or the second one?

All perfect praise be to Allah,The Lord of The Worlds                                                                                                                                                                    The time of Fajr prayer starts after the second Adhan. And Allah Knows Best.

When is it Sunnah to slaughter the 'aqīqah?