Is it permissible for a woman who is in her confinement to fast upon seeing blood signs that are neither black, nor red ?
Brownish, reddish, and yellowish discharges are all considered confinement until there is no colored discharge which marks the end of confinement. Moreover, the aforesaid woman is prohibited from fasting until she attains ritual purity. And Allah Knows Best.
Does an internal examination require ghusl?
Ghusl isn't required for making internal examination; however, it invalidates one's fasting. And Allah Knows Best.
Is Aqeeqah due on the miscarried embryo?
No Aqeeqah is due on the miscarried embryo.
When is Istikharah (guidance prayer) Du`a offered, before Tasleem (saying asalamu alikum warahmatu allah wa barakatuh to end the prayer), or after it?
It is offered after Tasleem.