How long is the` Iddah (waiting period) of a divorced woman?

The waiting period (iddah) of a divorced woman who is pregnant is until she gives birth. If she is of menstruating age, her waiting period is three menstrual cycles. If she does not menstruate, her waiting period is three months. And Allah Knows Best.

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.

Is the patient who is on dialysis twice a week obliged to observe fasting? Is his fasting on the days he is on dialysis considered valid or not?

Praise be to Allah the Lord of the Worlds.

Dialysis invalidates fasting because of the nutritional fluid used in that procedure, as pointed by specialized doctors, in addition to causing bodies to enter the stomach. Therefore, it is recommended that the patient who had undergone dialysis to observe fasting for the rest of that day, if he is able to, and make up for it after Ramadan. However, if the doctor told him that fasting, at any time, would damage his health, then he must break his fast and pay a ransom against every missed day. We pray that Allah blesses this person with recovery. And Allah The Almighty Knows Best.

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.