Wa `alaykum As-Salamu wa Rahmatullahi wa Barakatuh

The `Iddah must be observed from the time the Khul` is valid, but it is only one month not three months as it was in the case narrated by Ibn `Abbas about the wife of Thabit Ibn Qais.

If the court is a civil court with no religious authority, then yes she must still do the Khul` by a Mufti. Divorce or Khul` is a religious practice and must be observed according to the Islamic Shari`ah. If in certain countries civil courts demand documentation for divorce, it does not substitute the need for the Islamic divorce or Khul`.

Almighty Allah knows best.