Is a divorce that occurred under duress and was tied to a condition that was breached considered Islamically valid, and is the wife's marriage thereafter valid?

Disputes are resolved in Sharia (definition) courts because of their ability to hear all parties and issue a ruling. A Mufti (definition) only hears from one party. If the divorce was written, it takes effect with the intention, and your intention was to effect the divorce, so it takes effect. The condition of not upsetting the children does not affect the divorce. One must consider religion and character when marrying and performing Istikhara, and raise children upon faith, goodness, and filial piety. It is not permissible for children to abandon their mother. It is the mother's right to marry, and her misconduct – assuming it exists – does not negate her right to her children's dutifulness towards her.