How does one know that Allah has accepted their repentance and forgiven them for a sin committed against themselves, especially with their intense remorse, inability to forgive themselves, and constant feeling of guilt? And will Allah answer their supplication if they ask Him for a worldly need unrelated to the sin?

Sincere repentance erases sins and makes the repentant person like one who has no sin. Allah answers the supplication of whomever He wills, and their abundance of sins does not prevent Him from doing so, just as He answered Iblis. However, some sins prevent the answering of supplication if their perpetrator does not repent from them with sincere repentance, especially if they are related to unlawful earnings in food, drink, and clothing. Lawful sustenance helps in the answering of supplication.