Is it permissible for the wife to request a divorce or khul’ in this situation according to Sharia, and what is the legitimate reason for that? And is khul’ treated as if we are still married or not? And what is the meaning of "fear of not observing the limits of Allah"?

Since you have accepted the khul' (divorce initiated by the wife), perhaps it is better for you given the entrenched disagreement and discord between you and them. The general rule is that it is not permissible for a woman to seek divorce or khul' except for a valid reason, such as the husband's dereliction of his duties towards her, or his harming her, or the deep-seated hatred she feels for him, making her fear that she will not be able to observe the limits set by Allah with him, such as her fear of his injustice and his disregard for her rights.