Yes, it is permissible to do so during the waiting period (the Iddah). If the husband agrees, they return to their bond of marriage. After the Iddah ends, she is not allowed to seek the return of the payment. However, they can get married again with a new marriage contract.

It is a condition in the khula divorce that the wife pays her husband in return for getting a divorce from him. Without such payment, a regular type of divorce must take place instead.

If the wife hates her husband in such a way that she does not fulfill her marital religious obligations with her husband or she threatens him in a serious way to stop fulfilling such duties, then it is permissible to divorce her by way of khula, by her paying him in exchange for him giving her the divorce. In this case, the husband cannot return to the marital relations during the waiting period (the iddah). In other cases, the divorce may not be by khula, like when they both agree to drop the husband’s financial obligations for the wife in exchange for him divorcing her. The divorce in this case is conventional, i.e. the man has the right to return to the marital relation during the waiting period.

All types of divorce, including khula and mubarat, are not valid unless there are qualified witnesses present, and the other conditions must also be fulfilled, such as the woman not being in her menstrual period and not having had sexual intercourse since the last menstrual period.

The khula divorce is not valid unless the woman hates her tie of marriage with her husband in such a way that her hatred leads her to not fulfilling her conjugal duties. Simply not liking him without reaching such a level of resentment does not suffice for bringing about the divorce by khula.
When all conditions are met, the woman can start the divorce formula by saying "khālaʾtuka ʿalā..." (I form a khula divorce for the amount of...), and she states the amount of money that she is willing to give him. The husband says afterwards with his intention of accepting the khulaʾ divorce, "Radhītu bi-dhālik" (I accept that).
If she does not hate him to the level of refusing his marital rights, the khula divorce cannot be established, and the normal revocable divorce can take place then, which does not become irrevocable unless the waiting period is completed.

The commencement of the iddah is from the time of the divorce until you see the third menstrual period. You must observe this iddah period if the marriage was consummated.

No, the khulaʾ divorce is irrevocable, so the husband is not allowed to return to the marriage just because he expresses his wish to do so. They marry each other with a new marriage contract. It is considered as one divorce.

The waiting period is three durations of purity between the menstruation periods, the first of which is when the divorce takes place. The waiting period ends when she begins her third menstrual period after the divorce. It is better that she waits until she finishes the third menstrual period before remarrying.

The divorce is valid. She owes him the agreed amount. He is allowed to demand it from her.

The marital bond between the couple returns without the need of a new marriage contract if the woman seek the return of her khula payment during the waiting period.

No, she can not without the return of the khula payment, but she and her previous husband can get married again by a new marriage contract, and if no other objection that prevents from the return to the marriage exists.