Couldn't the more NW features of Taymadic be because of the fact we attest it earlier compared to Arabic?
Well I mainly made the argument because I knew South Arabian didn't exactly respect the modern Saudi-Yemeni border, I know Najran on the Saudi border was under the Sabaeans too.
But what else can they realistically be if not one of those 2 or a branch close to them? In terms of chronological depth and the given geography I don't see how we could get a highly divergent language or branch within the time frame given between the divergence of Central or West Semitic and time we hear of Thamudic.
I imagine the southern spread happened with the Qedarites and Nabatean period.
1) Regarding Taymanitic, Kootstra and al-Jallad, who are the only scholars I know of who currently have delved deeply on the topic, do not seem to think so. By the way, the inscription found at al-Bayir in SE Jordan, which is now thought to be the earliest document of something that can be tentatively called "Arabic" is thought to be about the same age as the Taymanitic inscriptions - it is undated however, which makes this a reasonable guess and not a certainty.
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.
3) Everyone agrees that Thamudic scripts attest West Semitic languages, and most likely Central Semitic ones. These seem not to fit with either Arabic, NW Semitic, or Ancient South Arabian, and the corpus is too limited and poorly studied to say much, but still it is very likely that we are talking about Central Semitic languages that diverged sometime in the (possibly early) Bronze Age. The main problem seems to be our poor understanding of the inscriptions, and the limitedness of the info they convey (most texts seem to be very short sentences or simple proper names, other are undeciphered). Notably, a different picture occurs in Dhofar, where, outside well-understood examples of Ancient South Arabian, some short inscriptions in what has been called Sa'alkalic 1 and Sa'alkalic 2 have been found. These are very poorly studied, and even more poorly understood, but the educated guess is that they would attest languages ancestral to Modern South Arabian (which is generally thought to be West Semitic, but not Central Semitic, unlike
Ancient South Arabian which most scholars now believe to be Central Semitic).
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).
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.