What is the ruling on memorizing the Quran during menstruation, knowing that I do not read directly from the Quran but from a paper or from my phone?
It is not permissible to memorize the Quran during menstruation because memorization requires repetition, and a menstruating woman cannot touch the physical pages of the Quran. Similarly, if the Quranic verses appear on the screen of the phone, it is considered like touching the Quran. And Allah Knows Best.
What is the Islamic ruling on adoption, and what is the legal window through which a person can be kind to an orphan girl, attend to her needs and raise her as his own?
Kindness to an orphan entails a great reward from Allah, but adoption is prohibited and invalid. Therefore, it is permissible for the aforementioned person (for example) to make his brother`s wife breastfeed that girl so that he becomes her uncle by suckling provided that he informs her that she isn`t his biological daughter.
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.
Does ill-gotten money hinder the answering of the Du`a (supplication) and the means of subsistence?
Sins hinder the answering of the Du`a, and deprive the sinner from means of subsistence; whereas, acts of obedience facilitate making a living. Allah, The Almighty, Says (What means) : "And for those who fear God, He (ever) prepares a way out. And He provides for him from (sources) he never could imagine." [At-Talaq/2-3].