Is the use of official documents in which the mother's name was recorded in a way that does not conform to reality (with the intention of matching previous data) considered cheating and deception? And is money earned from work in which these documents were submitted considered unlawful, knowing that this alteration does not harm anyone's interest?

If there is a significant benefit to be gained or an unbearable harm to be averted, then there is no blame upon you for what you did, especially if the matter does not involve attributing lineage to other than the father, but rather deceiving others. Lying is permissible if it is for the purpose of achieving a benefit or averting a harm, and it is better to use equivocation (Tawriya). What you mentioned may fall under the category of allusions (Ma'areedh).