I say short term it leans Vader, long term Sauron
Who wins short term and long term, Sauron or Darth Vader.
Fyi battleground is on slopes of Mt Doom, no one exists besides Sauron, (and Illuvatur) Darth Vader is suddenly teleported.
Yep, and Darth Vader, unless he has been told about the ring, will A. grab it or B. leave it alone.
A. Becomes corrupted and eventually a wraith.
B. Sauron comes back with knowledge and catches Vader unprepared.
Darth vader is corruptible so he would be manipulated by Sauron and help him kill Frodo from afar and rule meddle earth empire style and the he would proceed to kill Sauron and challenge illuvatar where he would crush him but moments before being crushed by Vader's force choke he would be joined by all of the Ainour forces and he would finally be defeted and kind of a happy end
Vader based on power alone is stronger than Sauron. If they got into a fight, he would win (simply because force wielders are actually brokenly OP and he is insanely strong for a force wielder).
Sauron's only chance would be to out think Vader. He would have to either trick him that he is stronger, or convince Vader that they are allies and stab Vader in the back before Vader does it to him.
Theoretically, if someone actually used the full power of the Ring he could kill Sauron I think. The problem would be is that he would take his place...
Additionally, Vader could theoretically destroy the ring if he just used the force to yeet it into Mt. doom. But is this taking place in Middle Earth, or in the Star Wars Galaxy, or in a neutral universe?
If Vader could reach is max potential like Luke does in Legends (which he can't) it would be possible for him to achieve the state of Oneness with the Force and destroy the ring (probably, like 99%).
Also, is this Sauron pre-first age, First age, Second Age, or Third Age? All of this could effect how a fight would turn out.
Honestly, I think you are underrating Vader (and force users in general). Any high level force wielder can use force stasis to just freeze non force wielding opponents or just force crunch them. They are actually extremely broken in any fight outside their universe. I once saw a video that analyzed that Obi Wan would probably beat a certain fairly powerful anime character, and he is just a mid tier jedi.
That being said, Vader would most likely beat balrogs, and probably in most cases most of the Valar too.
Unless, that is, we are talking maxed Valar, as in, before they expended any of their power in the world. I believe that Melkor before giving a bunch of his power to Ungoliant would beat Vader 90%+ of the time.
As for the other Valar, they would win if Vader went into the fight underestimating them. If he started by going all out, they would probably need to team. The Force is just that broken.
Note that when I am saying Vader would win I am not saying that he would destroy the Valar. It is my belief that killing and of them would just weaken them and allow them to escape and regain their strength over time, or cause other problems for Vader in spirit form. I think this is the reason why the Valar never killed Morgoth because they knew it would just be letting him walk free (even though he would be severely weakened.)
Legends Luke could beat all of them if he went all out as when he is in the state of Oneness he quite literally has unlimited power.
We're using Darth Vader from episodes 1-6, and we're using Sauron right before he fights Isildur at the end of the Second Age.