Did Sauron survive?

Question

According to Tolkien himself Sauron had a physical form in the third age:

...in a tale which allows the incarnation of great spirits in a physical and destructible form their power must be far greater when actually physically present. ... Sauron should be thought of as very terrible. The form that he took was that of a man of more than human stature, but not gigantic. ~The Letters of J. R. R. Tolkien

The question that I'm begging to ask is, does he survive his ring being destroyed? Is he seen dying? Or is he simply not seen at all?

Answer 1

Sauron himself was not seen in the movie, only his Great Eye. As to his destruction: from the chapter The Last Debate from The Return of the King Gandalf says of the One Ring

"If it is destroyed, then he will fall; and his fall will be so low that none can foresee his arising ever again. For he will lose the best part of the strength that was native to him in his beginning, and all that was made or begun with that power will crumble, and he will be maimed for ever, becoming a mere spirit of malice that gnaws itself in the shadows, but cannot again grow or take shape. And so a great evil of this world will be removed."

Sauron does survive the destruction of the ring but he is powerless to take action in Middle Earth ever again. However, in the original ending of The Silmarillion Tolkien wrote that Sauron would arise during the Dagor Dagorath or "Last Battle and Day of Doom" when his former master Morgoth breaks the Door of Night and destroys the sun and moon.

Answer 2

Sauron continued to exist

In a 1968 interview, when discussing if he had met any Sauron characters in real life, Tolkien mentions that he "wouldn’t have liked to meet Sauron when his Ring had been destroyed", thus showing that he felt Sauron still existed.

Of course, there’s another thing, yes, yes, that is a power, yes, yes. People who’ve met some of the great characters [get] a sense of something which is nothing to do with them or their intellectual power. Yes. I wouldn’t have liked to meet Sauron when his Ring had been destroyed, very little exist[ed?]. [But] one does meet people like that. Fortunately the people I’ve met with this extraordinary will-power, which is quite unrelated to [any?] ideas, have nothing. Fortunately when I’ve met them they were fortunately not people with any particular desires, ambition or even sufficient intellect [?].
_{Lee, Stuart D. ""Tolkien in Oxford" (BBC, 1968): A Reconstruction." Tolkien Studies, vol. 15, 2018}

This confirms Gandalf's speculation:

If he regains it, your valour is vain, and his victory will be swift and complete: so complete that none can foresee the end of it while this world lasts. If it is destroyed, then he will fall; and his fall will be so low that none can foresee his arising ever again. For he will lose the best part of the strength that was native to him in his beginning, and all that was made or begun with that power will crumble, and he will be maimed for ever, becoming a mere spirit of malice that gnaws itself in the shadows, but cannot again grow or take shape. And so a great evil of this world will be removed.
_{The Lord of the Rings - Book V Chapter 9 - "The Last Debate"}

Answer 3

Sauron's spirit or essence survived, he no longer has or will have the strength to do evil again unless his entire essence was sent to Mandos in which case when the hall doors open all bets are off.

Morgoth/Melkor the Ainu that corrupted him still existed, though and he eventually was to return to face Tulkas (Thor like Demi god, a Vala), Fëanor (creator of the Silmarils), and Turin Turumbar (Herculean figure of the first age).

Turin, and his black sword Gurthang were to end Melkor's existence and with him all evil, but this was literally at the end of time.