Does the Khulʿ take place without informing the husband, asking him, and hearing from him, if the wife appoints a lawyer and claims to have obtained a fatwa for Khulʿ due to the husband's suspension of her, and she returns the dowry to him?

As long as you have not divorced or khula’d your wife, she remains under your marital bond. The fact that she obtained a fatwa for khula’ does not result in khula’ or divorce. Your mere acceptance of the money offered by her for khula’ does not constitute khula’. Khula’ is only valid through offer and acceptance between the spouses, or by the ruling of a Muslim judge, or what takes its place in countries that do not have Sharia courts.