Faraaidh, Sunnats and Waajibaat of Salaah

Muqtadi saying the takbeer when moving to different postures

Q: Is saying transitional takbeer (Allahu Akbar) when going to ruku, sujood, qiyaam obligatory upon the individual praying behind an imam or can he just follow the imam.

I have performed many prayers behind the imam and I thought I wasn’t obligated to say the takbeer besides takbeer al ihram. Would my previous prayers be invalid otherwise even if I were unaware? 

Moving one's lips while reciting duas in salaah

Q: I heard from a shaik that one should move his lips while reciting azkaar in salaah or in general, and he gave the example of "one will not get sin if they think of sinning except they say or act on it. and the shaik said that so similar the good deeds!"

a. Once I read in a hadith that the Prophet (Sallallahu Alaihi Wasallam) never moved his lips in Salatul Asr. (I was not aware of the authenticity of that and I also remember that I read it from Sahih Muslim/Bukhari). 

b. I heard from another shaik that if someone around you recites anything only by moving his lips or some phrase with a sound and some without, then he said that one should warn such a person to stop it or read it clearly and loudly, if not then we have to tell them that "you are a witch person and will reveal it to everyone if you don't, (because it is an act of witches or magicians, the shaik said)" now which is correct and in which situations?

Sunnah method of Salaah

Q: When you perform salaah, when doing takbeer to start, should we look up or down? And after raising from rukuand saying "samirallahhulimanhamida", some says you have to say "allahumma rabbanawakalhamd" and others only "rabbana lakalhamd"? Which one is correct according to sunnah? And is it obligatory to say it? If i did not say it, are all my prayers invalid?