Are the prayers and Hajj of a woman valid, knowing that she doubted her purity before the Fajr prayer and the Ifadah circumambulation after hearing a sound she did not pay attention to, and then intercourse occurred thereafter? And what are the consequences for her?

If you doubt the invalidation of your Wudu, your prayer and circumambulation (Tawaf (definition)) are valid because certainty is not removed by doubt. This applies to both doubt and strong presumption. However, if you are certain that your Wudu has been invalidated, your prayer is invalid, and you must repeat it. As for Tawaf, if you are certain that your Wudu has been invalidated, the majority of scholars stipulate purity for its validity; therefore, your Tawaf is invalid, and you must return to Makkah to repeat the Tawaf Al-Ifadah. If sexual intercourse occurs after the first 'Tahlul' (partial release from Ihram (definition)), it does not invalidate the Hajj, but a sheep is required as expiation. Some scholars hold that purity for Tawaf is merely recommended or an obligation that can be compensated by an offering (dam), and this offers more flexibility.