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.
What should a woman who delayed making up for missed fast, due to menstruation, till the start of the next Ramadan?
Whosoever broke fasting in Ramadan, and was able to make up for it, but didn`t until the next Ramadan started, is obliged to make up for the missed days, and to pay the ransom as well. However, if he/she wasn`t able to before next Ramadhaan due to an excuse, then he/she should only make up for the missed days. And Allah Knows Best.
Does the date for menstrual menses change, and how is the increase calculated?
If the bleeding lasts no less than (24) hours, but doesn’t exceed (15) days, and the purity interval between the two menses is no less than (15) days, then it is menstruation even if the date of menses changes, or its duration increases.
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.