Is it obligatory for a wife who has been abandoned for many years to observe the waiting period (iddah) after her husband passes away?
If her husband passes away, the wife is required to observe the waiting period (Iddah), even if he had abandoned her, because she is his wife and inherits from him, so she must observe the iddah after his death. And Allah Knows Best.
What should I do in order to wake up for Fajr (dawn) prayer?
All perfect praise be to Allah, The Lord of The Worlds What should I do in order to wake up for Fajr (dawn) prayer? Answer: Some of the acts that aid in offering Fajr prayer are: 1- Going to bed early. 2- Willingness to wake up. 3- Reciting Al-Korsi verse from Chapter Al-Baqarah. 4- Taking refuge in Allah from the cursed devil, and making ablution immediatley after being awake, but it is preferable to ask someone to help in this regard to perform the Fajr prayer. And Allah Knows Best.
Is it permissible for the guardian, or the father of the child to hit the latter in order to teach him/her good manners?
It is permissible for the guardian to hit his/her child for teaching them good manners once they become ten years old provided that it is done mildly, harmlessly and without intimidation. Actually, such punishment is to be employed gradually i.e. after: asking, warning, and scolding. As for the teacher, he isn`t allowed to hit the student unless the latter`s father approves of it, and it is done according to necessity without intimidation, or harm.
Is it permissible for a Muslim woman to wear trousers at home?
Wearing trousers at home isn't prohibited so long as non-Mahrams do not see her. And Allah Knows Best.