As for the awareness of Stalin having another name, I can't actually say for certain (I don't recall ever seeing it mentioned, honestly), but I would lean towards it being known. After all his oldest son Yakov used Stalin's "original" name Dzhugashvili (to use the Russian transliteration of the Georgian ჯუღასშვილი; transliterating to English gives Jughashvili) when he enlisted in the Red Army in 1941 (his second son Vasili and daughter Svetlana used Stalin). That Stalin was an ethnic Georgian was also well-known, and he famously spoke Russian with a distinctive Georgian accent, and as Stalin is derived from a Russian word (сталь, steel), it would not be hard to figure that out.

As for previous names, Stalin did go through quite a few. When he first started out as a revolutionary he was known simply as Soso, which is nothing more than the Georgian diminutive of his name, Ioseb (Joseph); this gets a lot more attention in the West than it should, as it's very common for Georgians to adopt names like this, and is the same as if he called himself "Joe".

As for nom de guerre's, Stalin's most prominent one in the underground would be Koba. This he adopted from a novel by the Georgian author Alexander Kazbegi, The Patricide. Described as a Robin Hood-type story set in 19th century Georgia, one of the main characters is a bandit-type fellow named Koba. Stalin apparently really enjoyed the book and saw himself in a similar role, so for a while was known as Koba (he utilized a variety of surnames with it, including "Besosvhili" and "Vissarionovich", which mean's "son of Beso/Vissarion" in Georgian and Russian, the latter being the Russian version of the name of his father) but certainly not "Stalin" at this time).

The first use of the name "Stalin" came in 1913 with the publication of the article that naturally made Stalin a known name in Bolshevik circles: Marxism and the National Question. Stalin had been asked by Lenin to write the article in order to clarify the Bolshevik stance on nationalism, which was a major source of debate at the time. Without getting bogged down here, I'll simply note Stalin's definition of a nation, which he said "is a historically constituted, stable community of people, formed on the basis of a common language, territory, economic life, and psychological make-up manifested in a common culture." This article, which received strong approval from Lenin and solidified Stalin as the authority on the nationality question, was signed K. Stalin (the K being for Koba). It was not the first time he'd used the name Stalin (a few minor articles had that byline earlier), but it was the most prominent example of it, and soon after he was using it exclusively, and would drop the pseudonym Koba as well.

Lastly, I'll link an answer I wrote a while ago about Bolshevik name changes. It was a common thing for early Bolsheviks to do, in part as a means to hide their identity from the tsarist authorities. Stalin was of course one of those people, but like the others he was not very successful at it (he was arrested multiple times, and was in Siberian exile when Nicholas II abdicated in 1917).

If you're looking for a detailed look at the names of Stalin, I'd suggest going for one of the biographies that explores his earlier years: Simon Sebag Montefiore's Young Stalin (2007) is probably the most accessible (and is indeed a great read), while Stephen Kotkin's Stalin: Volume I: Paradoxes of Power, 1878-1928 (2014) is a lot more thorough (as it is about everything even remotely related to Stalin). However as Stalin was heavily influenced by Georgian literature, I'm more partial to Ronald Grigor Suny's Stalin: Passage to Revolution (2020). It's also a huge book (900+ pages), but unlike Montefiore and Kotkin, Suny is familiar with Georgian culture and history, and actually speaks the language himself (no easy feat). I'm also an unabashed Suny fan, so take that for what it's worth.