Have I committed a sin by preventing my wife from going to the hospital at the beginning of her pregnancy's critical stage, which led to the death of the twins after premature birth? And has the wife committed a sin by disobeying me and being examined by the doctor, knowing that I had ordered her not to consent to examination by male doctors?

The default is for a man to be treated by a man, and a woman by a woman, unless there is a necessity, such as the absence of a competent physician of the same gender. In such a case, it is permissible for a woman to be treated by a male doctor and for a man to be treated by a female doctor, to preserve life and avert harm. Preventing a wife from receiving treatment in a dangerous situation when there are no competent female doctors is not permissible Islamically. The husband must accompany her and be present during the examination, or another woman should accompany her to prevent seclusion and temptation. What happened is considered a forgivable error if the motive was caution and not an intent to cause harm. Repentance, seeking forgiveness, and consulting scholars are necessary. If the cause of the fetal death was preventing the wife from treatment, then the husband owes two expiations (freeing two slaves or fasting for four consecutive months), and his 'aqilah (male relatives on his father's side) owe two blood moneys to be paid to the heirs of the two children.