Is the marriage that took place between the girl and her fiancé—who lied to her father, and the girl claimed that her father was preventing her from marrying—invalid, and does it necessitate applying the حد الزنا (hadd punishment for unlawful sexual intercourse) to them? And what is the status of their children if they conceive?

If this incident occurred before a Shar'i judge, then your marriage is valid, and the children are attributed to the man. Even if the marriage were presumed invalid due to the absence of a guardian or a judge's ruling, it is still a "marriage of شبهة (doubt/semblance)" through which punishment (حد) is averted and the child is attributed to his father. This does not fall under the category of Zina (unlawful sexual intercourse) or the implementation of its prescribed punishment.