Is the husband sinful for not speaking to his wife or having intercourse with her after strained relations between them and the intervention of her family, and is he obligated to pay for the cost of the orthodontic treatment his wife underwent without his knowledge?

If what you mentioned about the father of the wife’s actions is true, then he acted wrongly, and it was incumbent upon him to verify the facts. The intervention of family members should be for reconciliation. As for the wife, the husband has no right to abandon her or deny her conjugal rights if she is upright, and he should not hold her accountable for the misdeeds of others. He should communicate with her to ensure a happy life built on mutual respect and the fulfillment of rights, and she should assist him in honoring his relatives. They must both appreciate the blessing of children and prepare a righteous environment for the children to grow up in, for it contains the good of this world and the hereafter, and because a righteous child will pray for them after their death. If the wife needs to straighten her teeth to remove a defect, it is part of good companionship for him to help her to the best of his ability.