Was the husband obligated to fast as expiation for accidental killing on behalf of his two brothers who died in an accident for which he was not responsible, and can the wife fast on his behalf after his death?

If the husband did not cause the death of his two brothers, then no expiation is incumbent upon him, because expiation is only incumbent upon the killer. If there is no proof of his involvement, then the original ruling is his innocence. However, if you are certain of his involvement through transgression or negligence, then the preponderant opinion is the permissibility, rather the recommendation, of fasting on his behalf, due to the generality of the Prophet's (peace be upon him) saying: "Whoever dies and owes fasts, his guardian should fast on his behalf." And the guardian is a close relative.