Is her father's going to court to testify on behalf of a man he does not know, so that man can obtain citizenship, considered a false testimony, and does this entail a religious sin for him? And what is legislated for her to absolve him of this if that is the case?

The sanctity of avoiding false testimony is not hidden. Allah has commanded us to shun it, and the Prophet, peace be upon him, mentioned that it is among the greatest major sins. It is not permissible to resort to false testimony except out of a recognized, unavoidable necessity, and what was mentioned does not constitute a necessity that permits it. The man who bore false witness is considered a sinner, and his children should increase their prayers, seek forgiveness, and give charity on his behalf.