I wrote on an answer to a similar question less than a week ago (link here), but will try and expand a bit to address your other questions.

I would say that no, ethnic Russians would not have considered Stalin Russian, for the simple fact that he was both: ethnically not Russian, and that the idea of a Russian citizenship did not exist during his time in power.

The first part is hopefully self-evident (as a Georgian he was obviously not Russian, and he never denied his Georgian heritage), while the second may need a bit of elaboration. While it's common to refer to the Soviet Union as "Russia" and the Soviet people as "Russians", something that was done during the existence of the USSR and after, the two are not synonymous. While the Soviet Union did control nearly the entire territory of the former Russian Empire, and while Russians were both the most numerous and dominant nationality in the USSR, it was not exclusively a Russian state. This is of course most apparent in the figure of Stalin, but after the 1930s there was a concentrated effort to establish the idea of the "Sovetskiy narod" (Soviet person). This was not a Russian, or a Georgian, or a Tajik, and while they may speak Russian as the dominant language at work and in official documents, they likely also spoke their native language. Distinct Soviet customs were established, alongside the local ones, and the idea was that no one people was above the other. This fit into Marxist theory, which saw nationalism as a product of the bourgeoisie and meant to divide the workers, rather than have them unite and fight the real enemies (this is seen in the slogan "Workers of the world, unite").

That all said, how does Stalin and his self-description of a "Russified Georgian" come to play here? Stalin of course was born and raised in Georgia, which was part of the Russian Empire until he was nearly 40. Growing up in the small town of Gori (about 60km from the capital Tbilisi), Stalin attended a church-backed school there, and moved to Tbilisi to attend seminary school when he was a teenager. Life in Gori was mainly conducted in Georgian and most of the local business would have been as well, with only dealings with government officials being in Russian. School instruction was also in Russian, though Stalin apparently did not learn much before he started classes there. At the seminary classes were also in Russian, though the public still spoke Georgian, and again only government officials used Russian on a frequent basis. In short, Georgian was still the dominant language of the country, and only those tied to the government used Russian at all.

Stalin learned Russian in school, though he was very much committed to his Georgian identity at this time. This era had seen a major revival in Georgian identity, especially among intellectuals like Ilya Chavchavadze and Giorgi Tsereteli, who promoted the use of Georgian as a literary language. Stalin followed this idea for a time, though as he developed his Marxist leanings and became a socialist, he felt that Georgia was not developed enough to fully embrace socialism, and that only through Russia (which itself was seen as backwards by contemporary European socialists; even Marx himself dismissed Russia) could the goals of socialism be implemented. As Ronald Grigor Suny has argued in his recent biography of Stalin (see below), he was a realist and a pragmatist, but he was much more than that, and as he developed, "[h]is conviction that Russian culture and society were more modern, more proletarian, and therefore superior to the cultures of the peoples of the periphery, particularly those of the southern and eastern borderlands [ie. Georgians, - KM], grew stronger in the postrevolutionary years of civil war." (pp. 691-692)

As to why Russians were "so willing" to allow someone like Stalin, a non-Russian, to rule them, that would arguably be due to his strong drive, and the ethnic diversity of the early Bolsheviks. To address the second point first, the Bolsheviks were a diverse crowd: aside from Stalin, the leadership had Trotsky (who was Jewish), Anastas Mikoyan (Armenian), Feliks Dzerzhinsky (Polish), and even Lenin (who possibly had Kalymk or Chuvash ancestry, possibly Jewish, too). There are more, but I think that gives the idea that they were not just filled with Russians (and arguably attracted non-Russians above all). As to Stalin’s drive: when Lenin started to get sick, Stalin worked to manoeuver himself into position to succeed, and managed to build a support base that would both allow him to take over, and ensure that Trotsky (the logical successor) was discredited and ultimately expelled. He further did this to consolidate his position in the 1920s and early 1930s, so that by 1929 he was the uncontested leader of the Party and by default the Soviet Union.

Lastly, I’ll mention that yes, Stalin was well-known to be Georgian. As you noted he spoke Russian with a thick Georgian accent, and would have had no reason to hide his background, though after he took power he rarely visited Georgia (he did spend time in Abkhazia during the summers, and nearby Sochi), and never explicitly favoured Georgians (Lavrenti Beria being the exception; Beria himself built up a strong network in Georgia though, and developed a clique of ethnic Mingrelians that supported him).

I hope that brings some clarity to the question, but I’m happy to further elaborate.

And as for some sources:

• Stalin: Passage to Revolution by Ronald Grigor Suny (2020). This was only released in October, and is a great book. I’m an unabashed fan of Suny so will admit to being biased, but Suny is one of the few Western historians who speaks Georgian, so made extensive use of Georgian-language sources here. The book looks at Stalin’s life up to 1917, and asks similar questions to you: how did someone from a poor Georgian family end up in a position to lead the USSR? It’s a huge book (700 pages, plus citations and bibliography), but if you want to understand how Stalin got to where he is, read this.

• Stalin, Volume 1: Paradoxes of Power, 1878-1928 (2014) and Stalin: Waiting for Hitler, 1929-1941 (2017) by Stephen Kotkin (a third volume is planned still). These are also huge books, about 800 pages each of text, and go far beyond just Stalin but encapsulate the world he was living in. Kotkin does a great job of going over the political side of things, and gets really into the details of the political struggles Stalin faced.

• "Stalin, Man of the Borderlands" in The American Historical Review Vol. 106, No. 5 (Dec. 2001), pp. 1651-1691 by Alfred J. Rieber. This article was later revised and used as the first chapter of Rieber’s book Stalin and the Struggle for Supremacy in Eurasia (2015), but this is also a key look at the early life and development of Stalin, and how his surroundings influenced him.