Is it permissible for a mother to file a lawsuit against her ex-husband who incites their children against her, takes them to dangerous places, and teaches them obscene words, knowing that the children are in her custody?

If the problem between the spouses can be resolved through the intervention of conciliators from both families to ensure a familial solution without resorting to courts, then that is preferable; due to its positive impact on the children and the reputation of both families. If that does not succeed, then your custody of the children is forfeited upon your marriage to someone other than their father, and it transfers to your mother. The father cannot be prevented from visiting his children or them visiting him, because prevention leads to severing kinship ties. The father has the right to take his children to recreational places, as long as there is no fear of definite harm to them in those places.