Is it permissible for a wife to return to her ex-husband after two divorces without obtaining a divorce certificate, and is a new marriage contract required for this?

Anger does not prevent the occurrence of divorce as long as the person is aware of what he is saying. So, if your husband divorced you while being aware and did not take you back until your iddah (definition) (waiting period) expired, then you have become irrevocably separated from him. He is not permitted to take you back except with a new contract if the divorce was revocable. However, if the divorce has reached three (divorces), you are not lawful for him unless you marry another man out of genuine desire, and then he divorces you after consummation or dies, and your iddah from him expires. The issuance or non-issuance of a divorce certificate does not affect the occurrence of the divorce or the permissibility of remarrying.