Can I start fasting based on the moon sighting of another country?
You must follow the moon sighting of the country you are in.
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.
Does using a respiratory device (oxygen) affect the validity of fasting?
Using an oxygen device does not affect the validity of fasting because oxygen is a gas with no physical substance.
However, if medicinal substances with physical particles are added to the oxygen, its use would break the fast, as it enters the body cavity through a natural passage.
A person who has recently embraced Islam is afraid of undergoing circumcision (khitān) — what should he do?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah ﷺ.
If this person would be harmed by undergoing circumcision, based on the assessment of a trustworthy physician, then he is not obligated to undergo it. However, if the matter is simply a fear of the procedure itself, then it may be postponed until he feels at ease and settled about it, provided this does not lead him to become averse to Islam — while still taking care to remove any impurity from the concealed area [beneath the foreskin]. And Allah, the Exalted, knows best.