As a bit of a sidenote, how does the fact Yemen still likely spoke South Arabian fit with the idea that the Ghassanids and Lakhmids were of Yemeni origin while being Arab? Something doesn't fit for me, especially considering the mode of migration seems to be with North Arabia as the point of origin and going in all directions rather than coming from the extreme south.2) Correct. Ancient South Arabian is documented into SW modern South Arabia and was probably spoken at Najran in the Iron Age - and likely later. Arabic spread there, likely in the Roman period and certainly by Late Antiquity - some of the earliest attestations of Arabic script are from there, dated fifth century CE. Earlier inscriptions from the area are either in South Arabian or in so-called Thamudic F (also called Himaic) but show admixture from likely some form of Arabic.
It's noot related to Central Semitic in Arabia but I heard arguments about a Cushitic substratum in Modern South Arabia, which is interesting given it would have an extremely remote time depth if true.The linguistic landscape of Arabia before the general spread of Arabic seems to have been rather diverse, even if all the languages involved likely belonged to the same branch, and most to the same subbranch within it, of Semitic, and little direct evidence for non-Semitic languages anywhere has been found (indirect evidence in substrate placenames has been proposed).
I found this:4) It may have started a bit earlier, but mostly yes, the spread of Arabic into the whole Peninsula is thought to have happened largely in the Qedarite and Nabatean periods, with the exception of Greater Yemen (and perhaps Oman, about which we have nearly no evidence at all). The central parts of the peninsula yield a lot of written material tentatively dated to Hellenistic and Roman periods (mostly Thamudic, but some showing traces of actual Arabic), but very little of it is firmly datable so you have much room for hypotheses.