Is the `Iddah (waiting period) of the divorced effective from the time the divorce took place, or from the time of registering it at the court?
It is considered effective from the very moment her divorce took place, but not from the time of registering it at the court. And Allah Knows Best.
Is the father a Mahram (Non-marriageable) to his son`s mother-in-law?
The father is a non-Mahram (Marriageable) to his son`s mother-in-law, so it is impermissible for them to look at each other, or to have a seclusion (Khalwah).
Does an internal examination require ghusl?
Ghusl isn't required for making internal examination; however, it invalidates one's fasting. And Allah Knows Best.
I missed fasting 30 days of the obligatory fast within the last 6 years and 20 days. How do I fast? And what should I do?
Missed fasting days must be made up, and it is permissible to make them up on non-consecutive days, as continuity is not required for making up missed fasts. Additionally, expiation (Fidya) must be given: feeding one needy person for each day if the makeup is delayed for one year. If the delay extends to two years, Fidya must be given to two needy people per day, and so on. And Allah Knows Best.