Can a woman perform I‘tikaf in her home?
No, a woman's I‘tikaf is not valid if performed at home, because her home is not a mosque.
What is the ruling on a woman who takes medication to delay menstruation for fasting?
If a woman takes medication (to delay menstrual period) and does not experience menstruation, her fast is valid. However, she is not advised to do so unless there is a necessity.
If the medication harms her, even potentially, it is forbidden (haram) for her to take it.
What is the ruling on eating and drinking at night after making the intention? Is it necessary to renew the intention?
Eating and drinking at night, even after making the intention (for the next day), does not affect the fast, and it is not necessary to renew the intention after eating and drinking.
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.