Is it permissible for a Muslim physician to specialize in gynecology obstetrics, and what is the ruling on working as such?

Some fields of knowledge are an individual duty while others are a collective one, and specializing in gynecology obstetrics is a collective duty. However, if there were female physicians to treat women, then there is no need for a male gynecology obstetrics specialist to look at women`s Awrahs (private parts) except in necessary situations since the jurisprudential maxim says:” Necessity must only be assessed and answered proportionately.” And Allah Knows Best.

Is it permissible for a person who hasn`t made ablution to call for prayer (Adhan)?

All perfect praise be to Allah, The Lord of The Worlds.                                                                                                                                                                      It is disliked for a person with a minor impurity to call for prayer, but if he does , then it`s valid and sufficient. And Allah Knows Best.

I am a woman, and my natural menstrual cycle lasts six days. In the evening, I perform the testimony (Tashahhud) for myself, and then I engage in sexual intercourse with my husband. After intercourse, I experience bleeding, either immediately or the next day. Is my intercourse with my husband during this time forbidden?

 

Is it permissible to slaughter one animal as a sacrifice and an Aqeeqah (the sheep slaughtered on the seventh day from the child`s birth)?

It is impermissible to do so since each of them is slaughtered for a different reason.