What is the ruling on someone who curses the religion or commits an act of disbelief during the day in Ramadan?

Whoever apostatizes (leaves Islam) while fasting, their fast is invalid. Cursing the religion is an act of apostasy (may Allah protect us from it). Such a person must return to Islam by pronouncing the Shahadah (testimony of faith), seek Allah’s forgiveness, refrain from eating and drinking for the rest of the day, and make up for that day’s fast later.

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.

If someone regularly fasts the six days of Shawwal, are they obligated to fast them every year?

A person is not obligated to fast the six days of Shawwal every year, even if they have made it a habit. However, someone who regularly performs a good deed should not abandon it as long as they are able.

Is a woman's prayer performed while following an Imam whose prayer is being broadcasted via the television valid, and does she receive the reward of congregational prayer?