The woman would struggle to support the children alone as a single parent with only her income to depend on. The children would also be traumatized after their father leaving them and they would most likely develop adaptation issues as they grow. The woman would most likely seek to get married again in order to get companionship and financial assistance in taking care of the children.
Depends on the situation. Plenty of couples have amicable divorces without issues, and there is little to no struggle on the mother or the children's parts. On the other hand, a nasty divorce or sudden disappearance is a very emotionally charged situation and will take some sort of emotional toll. Whether or not there will be financial struggle or long-term problems is, again, dependent on the situation and the individuals in question.