Is it permissible for the man to go out wearing shorts?
It is permissible as long as his Awrah(private part), which is between his navel and knee, is covered. However, the regular wear traditions and modesty are to be observed.
My husband told me that he concluded our marriage with a fake name that belongs to another person, because he was sentenced. Nowadays, he brought an ID, passport and birth certificate with the his current name. What is the ruling on being married to him?
praise be to Allah, The Lord of The Worlds, and may His peace and blessings be upon our Prophet Muhammad and upon all of his family and companions.
As regards being married to the person mentioned in your question: The marriage is valid so long as he was the same person your guardian concluded the marriage with, even if he changed his name i.e. if your marriage contract was concluded with the same person, since what counts regarding marriage is the persons not names. And Allah Knows Best.
Is it permissible for the wife to give the Zakah(obligatory charity) of her money to her poor husband?
Yes, it is permissible for the wife to give from the Zakah of her money to her poor husband.
Who is required to fast?
Any adult, sane, and hardship enduring Muslim person is required to fast unless there were prohibiting conditions such as menstruation and confinement.