What is the ruling on entering the toilet with something containing the remembrance of Allah?
It is disliked to bring into the toilet anything containing a reminder of Allah, such as the Quran or a sacred name (like the name of Allah, Muhammad, etc.). If one does that, etiquette dictates that he conceal it so it is not visible while entering the toilet, either by cupping his hands over it, putting it in his pocket, or inside his shirt if it is a necklace or ring. And Allah the Almighty knows best.
What is the Islamic ruling on breaking the bones of the 'aqīqah?
All praise is due to Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
It is Sunnah not to break the bones of the 'aqīqah. Rather, each bone should be separated at its joint — as a good omen and expression of hope for the soundness and wholeness of the newborn's limbs. However, if one does break the bones, it is not considered disliked (makrūh) — it is simply regarded as contrary to what is preferable (khilāf al-awlā). And Allah Almighty knows best.
Is it permissible for a woman to offer the obligatory prayer after Adhaan (call for prayer), and before Iqamah (The announcement that the prayer is about to begin) in the mosque?
Yes, it is permissible for a woman to offer the obligatory prayer immediately after the Adhan and before the Iqamah in the mosque. And Allah Knows Best.
Should a person feel pleased, or have a virtuous vision after offering Istikhara (guidance prayer) in order to do what he/she had offered it for?
The result of the Istikhaarah is not necessarily that a person sees something (in his dream), or feels pleased, but the most important result of the Istikhaarah is whether a person is enabled to do a given matter or not.