Question: A man divorced his wife three times. Is it allowed for her to stay in the same house, for the sake of the children, if they avoid each other? Can she marry another man, get divorced, and then marry the first husband again?
Answer: She is allowed to stay in the house as long as they do not touch each other, look at each others ^awrah, show their ^awrah to one another, stay alone in a room without a 3rd person they would feel shy from, or anything else that is forbidden between strangers.
If she married another man and had intercourse in that marriage (this is a must), and was divorced from that man, then she can marry her 1st husband again. However, it is not allowed to make a marriage with the stated condition that it is for dissolving the 3 divorces made by the first husband. If the condition was not stated explicitly, then it is ok. Beware that the second husband cannot be forced to divorce her, i.e. he might choose to keep her.