Is purity a condition for prostration of recitation (Sujud al-Tilawah), and is it required to say "Allahu Akbar" when prostrating and rising from it (whether inside or outside of prayer)? What should be said during this prostration, and is the supplication mentioned for it authentic? Is it permissible to say the salutation (taslim) after it if it is performed outside of prayer?

Purity (Taharah (definition)) is not a condition for the prostration of recitation (Sujud (definition) al-Tilawah). There is no salutation (taslim (definition)) or takbir (uttering "Allahu Akbar") when rising from it, but one should say the takbir when prostrating for it. As for prostration within prayer (Salah (definition)), the takbir is obligatory when lowering and rising. The remembrance (dhikr) and supplications (du'a) that are prescribed for the prostration in prayer are also prescribed for it, such as: "O Allah, to You I have prostrated, in You I have believed, and to You I have submitted. My face has prostrated to the One who created it, fashioned it, and gave it hearing and sight by His might and power. Blessed is Allah, the best of creators," and "O Allah, record for me by it a reward with You, wipe away from me a burden by it, make it a treasure for me with You, and accept it from me as You accepted it from Your servant Dawud (peace be upon him)." The obligatory utterance is: "Glory be to my Lord, the Most High," and anything more is recommended. Sujud al-Tilawah is a Sunnah (definition) (recommended act), not an obligation.