Should the Zakah (obligatory charity) giver tell the poor recipient that this is the Zakah of his money?
No, he shouldn`t. But, the Zakah giver should make an intention in his heart that it is the Zakah of his money when paying it to the poor. And Allah Knows Best.
Is it permissible for a praying person to make supplication in each Sajdah (a prostration) of obligatory prayer?
It is permissible for a praying person, whether praying individually, or in congregation, to make supplication during Sujood (prostration) in obligatory, or voluntary prayers. However, it is disliked for the Imam to do so since he is required to spare the prayer performers any sort of hardship, unless he is leading a certain group who don`t mind him extending the prayer.
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.
My headcloth is causing me a terrible headache and severe shortsightedness. Is it permissible for me to take it off?
All perfect praise be to Allah, the Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallahu ‘alayhi wa sallam, is His slave and Messenger.
If the headcloth Khimar is harmful, it is sufficient to wear the Niqab (face-veil). However, with the passage of time you will get used to the Khimar and these symptoms will go away. And Allah The Almighty Knows Best.