What type of illness that permits breaking the fast in Ramadan?
It is an illness that, if one fasts, there is a fear it may lead to their death or cause unbearable hardship.
Which is better for a traveler: fasting or breaking the fast?
Fasting is better, unless it causes hardship—in that case, breaking the fast is better.
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.
Is it permissible to pray for mercy for deceased non-Muslims?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah.
Yes, it is permissible to pray for mercy upon them.
And Allah Almighty knows best.