Answer
(Fatwa: 1526/1307=L/1429)
(1) Yes, salatut-tasbih is proved by the Prophet (صلى الله عليه وسلم), he taught his uncle Hadhrat Abbas (رضى الله عنه) the method of performing this salah which is mentioned in Tirmizi Sharif vol. 1 p. 109.
Allamah Shami declared the Hadith of salatut-tasbih as Hasan, because it has been narrated through many ways: وحديثها حسن لكثرة طرقه (شامى 2:471)
Many virtues of salatut-tasbih have been mentioned in Hadith. One who offers it will get a lot of rewards.
Salam is a dua and greeting. It can be used also for people apart from the prophets. However, if somewhere the Shiahs are dominant and they suffix the names of Hadhrat Ali and Hadhrat Hussain with alaihis salam considering them imam masoom, then one should avoid it.
Allah (Subhana Wa Ta’ala) knows Best
Darul Ifta,
Darul Uloom Deoband