Is the Aqeeqa due on the newborn in case of dying almost a month after the delivery?
Offering Aqeeqa after a week of a newborn death is recommended.
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.
What is the expiation for perjury?
Perjury is forbidden and one of the major sins that require turning to Allah in repentance, seeking His forgiveness, giving back rights to whom they belong, or seeking their forgiveness, and expiating for that oath.
I`m a pious Muslim woman, but my husband isn`t, what should I do?
You should exercise patience, make supplication that Allah guides him to the straight path, and keep advising him kindly.