Is the sister considered sinful and does she owe expiation if she swore before the judge that she did not know that a certain person was her brother through rada'a (breastfeeding), knowing that she generally knew about the issue of rada'a but did not specifically know about that person, and she felt peace and tranquility after the oath, and that her wet nurse tried to keep her with the husband to confirm the fatwa of rada'a?

Istikhara (seeking guidance from God) is not a means of deriving legal rulings, nor can Islamic legal rulings be inferred from it. Rather, the way to learn about them is by asking specialized scholars, as stated in the Almighty's قول: "So ask the people of the Reminder if you do not know."

Likewise, Islamic legal rulings cannot be derived from visions or dreams.

As for the oath your sister took, it depends on the intention of the one who administered the oath: if the intent was that she did not know the man was her brother, then there is no sin upon her if she swore, because she is truthful. However, if the intent was that she did not know about the suckling relationship at all, then she is sinful for swearing to a falsehood.

There is no difference in swearing by the Quran whether one touches it or not; both are considered valid and weighty oaths in Islamic law.