@DKhantastic Q: Where was Ram born, Ayodhya or Kurukshetra?

A: Speaking in strict historical terms, we have no evidence that Rama was a real historical person, so, neither. As a matter of modern Hindu religious belief, the standard answer is that Ram was born in Ayodhya.

@apveng Q: Best Sanskrit and English editions of Valmiki's Ramayana?

A: The Goldmans/Pollock translation of the critical edition of the Ramayana is the best literal English rendering. The Clay editions of book 1-5, I believe, have transliterated Sanskrit en face.

@iwanderer Q: Is the story of Shambuka present in Valmiki's Ramayana?

A: Yes, it is in the Uttarakanda, volume 7 (many of whose episodes are of relatively late origin). It was known by at least the 7th century since it appears in Bhavabhuti's Uttararamacarita (possibly earlier; I've have to check). The episode appeared to disturb numerous later commentaries who added elements designed to soften the beheading, such as adding an element of divine approval or showing Rama hesitating.

@Spectatorkhan A: How many versions of the Ramayana exist?

A: Countless, and there are more all the time. It is a story always being retold. Even what we call Valmiki's Ramayana was a fluid text that existed in multiple versions.

AK Ramanujan began a famous essay on the Ramayana thus: "How many Ramayanas? Three hundred? Three thousand? At the end of some Ramayanas, a question is sometimes asked: How many Ramayans have there been? And there are stories that answer the question. Here is one."