Is the Prayer (Salat) established for both the Sunnah and the Fard (obligatory), or for the Fard only? And is it permissible to say the Takbir after the first Iqamah?

The Iqamah (call to commence prayer) is not prescribed for supererogatory prayers (whether regular Sunnah prayers or others). It is exclusively for the five obligatory prayers. So, if you intend to pray an obligatory prayer, you may make the Iqamah for it if you wish.