designs,
Imo the Christian "communities" which produced the NT writings, including the Gospels, can be traced back to a number of "roots" in and out (or on the fringe) of Judaism, both in Palestine and in the diaspora. Some of these "root" circles -- which may or may not have considered themselves "Christian" -- may have used the divine name in some form (in writing and/or orally, as a paleo-Hebrew transliteration requiring oral substitution, or in pronuncible Greek transliteration like Iaô): in our present state of knowledge we can't either prove it or rule it out. What remains is that the actual pronunciation or writing of the divine name in any form apparently played no part in the stage of "Christianity" which produced the earliest known Christian texts, although the notion of the "divine name" remained an important theological motif (cf. the Lord's Prayer which parallels the Pharisaic Qaddish in Matthew, neither of which requires pronouncing the name btw; the Christological uses of kurios in Paul, the "name" manifested by the Son and given to him in John, more loosely the absolute egô eimi... all those are the echoes of the theme of the divine name in the NT, but do not imply actual use of the divine name in any form at that stage).
Now when you ask, "When Jesus quotes the Torah in Matthew 4 did he speak in Aramaic, Greek, did he quote the Torah in comptemporary Hebrew of the day or in the dialect of say King David's day?" -- you forget to ask: did he? I mean, you seem to simply take for granted that a late 1st-century Greek narrative (in that case, a rabbinic-like exchange of prooftexts with the devil, with no witnesses!) actually reflects something that happened to a Palestinian character in the first century: this assumption I find very questionable.