Does Allah make the sin forgotten for its doer and for those who became aware of it, upon granting pardon and forgiveness? And is it permissible for a Muslim to supplicate to Allah to make his sins forgotten?

What you have mentioned about your husband's and mother's reactions confirms that the mistakes you recall did not originate from you, and that they are merely whispers from Satan. Even if they did occur, your remorse and your husband's and mother's forgiveness are sufficient, for "the one who repents from sin is like one who has no sin." Do not let Satan toy with you, and there is no evidence that Allah makes the penitent forget their sin. There is no harm in asking Allah to make you forget your sins.