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.

Should a woman who broke her fast because of delivery make up for missed fasting days before the next Ramadhaan, and what is the expiation due on her in case she delayed making up for them ?

She should make up for missed fasting days before the start of next Ramadan if possible, but if she didn`t while being able to, then she is obliged to make up for them along with feeding a needy person for each delayed day of the missed fasting days. However, if she wasn`t able to make up for the missed fasting days before the start of next Ramadan, she has to fast a day for every day that she missed, and no ransom is due on her. And Allah Knows Best.

What should a praying person who doubts having offered one, or two prostrations in the third Rakah (unit of prayer) do?

Whoever has doubts about their prayer should consider what is less i.e. the above person should offer another Sajda (prostration) and complete his/her prayer, then offer Sujood As-Sahw (prostration of forgetfulness). And Allah Knows Best.

Someone asked me to pay off his debt on his behalf as a loan, without any compensation. When I went to the creditor, he told me that if I paid the full amount at once, rather than in installments, he would give me a certain discount. Is this permissible? And if he applies the discount, to whom does the deducted amount belong?