All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
It is not permissible for your friend to do the marriage contract in which both the husband and wife do not comply with the rights and obligations they have toward each other. This act is a kind of playing with the rulings of Islam and disrespecting them. Allaah says (what means): {These are the limits of Allaah, so do not transgress them. And whoever transgresses the limits of Allaah- it is those who are the wrongdoers [i.e. the unjust].}[Quran 2:229]
The need of this girl to get the legal status does not make this kind of marriage lawful.
However, if the marriage is carried out fulfilling all the conditions of a valid marriage, and it was free of the condition of not maintaining any relation between the married persons, i.e. the husband and wife, then the marriage is valid. Hence, all the results of a valid marriage take effect such as the lawfulness of having sexual intercourse, being in seclusion, and spending on the wife, etc. However, if the marriage contract included the conditions mentioned in the question, then the marriage is invalid.
As regards the money which the girl wants to give to your friend, it is forbidden money as it is compensation in return for a forbidden act.
Allaah Knows best.