Is it obligatory for the wife to offer an Udhiyah if her husband offers it on behalf of the family, and is it better for her to offer it in her name and then give charity, or to give the ram alive as charity, desiring the reward?

The Udhiyah (sacrifice) is a confirmed Sunnah for the one who can afford it. One Udhiyah is sufficient for the members of a household, and it fulfills the requirement for the rest of them. The reward is for the one who offers the sacrifice. It is a communal Sunnah (Sunnah Kifayah); if one person from a household makes the sacrifice, the obligation is lifted from the others. However, the reward is for the one who sheds the blood. Slaughtering the sacrificial animal and distributing it to the poor is better than giving its value in charity.