Thu'um is one of those interesting cases in which the idea was mentioned and discussed multiple times, and yet was still somewhat different in those references than it was when it featured as a central idea. As others have said, one does not need to be Dragonborn to learn the Thu'um; in its original use, in fact, there was no connection between Dragonborn and Thu'um at all, except that Tiber Septim happened to be a Dragonborn who had learned to use it (or claimed to, depending on your interpretation of the Heresy). Connecting them as Skyrim did was a design choice in order to make the player character's journey to hero-dom mirror Tiber Septim's own, at least as far as I can tell. This, of course, also led to several previous Thu'um users being labeled "Dragonborn" simply because they were important and therefore should be given a title just as important as Tiber's and your own (such as Wulfharth, the most famous of them). It should be noted, though, that the similar title "Ysmir" ("Dragon of the North") was used in the lore for both Wulfharth and Tiber (in fact, I believe they were the only two to hold the title until you gained it in Skyrim). The devs conflated the Ysmir title with Dragonborn, thereby giving you that title automatically regardless of our previous understanding of the term.
Other than the title Ysmir, though, there was little to nothing that connected dragons to the Thu'um prior to Skyrim. Certainly the dragon language of Skyrim did not exist. The previous stories relating to Thu'um said that Kyne was responsible for giving it to Men; the only brief mention of learning a shout through fighting a dragon came from the Five Songs of King Wulfharth, when he learned "what happens when you shake the dragon just so," or something along those lines (though one should note that Alduin doesn't die during this fight, so clearly Wulfharth didn't steal his soul in order to learn this shout). Although this never turned up in-game, it is interesting that Tiber Septim's Sword-Meeting with Cyrus the Restless features shouts that I'm fairly certain are Ehlnofex (if someone could confirm this, I'd be grateful). They are certainly not made up of the words of Skyrim's dragon language.
/rant
Everyone else has already answered your other questions better than I could.