I'm a bit confused as to the ordering of events leading to Sauron's demise at the end of the second age.
From the Silmarillion it appears that Sauron was hewn down by Gil-galad and Elendil, and afterwards the Ring was cut from his dead body's finger by Isildur. Is this the correct order of events? Or was Sauron weakened by Gil-galad and Elendil and then Isildur cut the ring from his living body causing it to be destroyed?
You're correct; the order of events based on the published texts is unclear.
In Of the Rings of Power and the Third Age we read the following words of Isildur:
This I will have as were-gild for my fathers death, and my brothers. Was it not I that dealt the Enemy his death-blow?
This is absolutely unambiguous; according to this text it was Isildur who killed Sauron. Yet, in the Fellowship of the Ring we read a different story:
It was Gil-galad, Elven-king and Elendil of Westernesse who overthrew Sauron, though they themselves perished in the deed; and Isildur Elendil's son cut the Ring from Sauron's hand and took it for his own.
This suggests that Sauron was killed by Gil-galad and Elendil, and that it was only after that that Isildur cut the Ring off.
Ultimately it seems as though we must accept that this was an event from very ancient history (over 3000 years before the Fellowship formed) and that, while there were people around who remembered it (specifically Elrond, who was there) the event itself is clouded by conflicting accounts.
According to The Silmarillion, Sauron was weakened after killing Gil-Galad and Elendil. That allowed Isildur to make the "Coup de Grace" of cutting off the Ring.
Peter Jackson's version is actually pretty close to what Tolkien wrote.
Here's the relevant passage from The Silmarillion:
Of the Rings of Power and the Third Age:
Then Gil-galad and Elendil passed into Mordor and encompassed the stronghold of Sauron; and they laid siege to it for seven years, and suffered grievous loss by fire and by the darts and bolts of the Enemy, and Sauron sent many sorties against them. There in the valley of Gorgoroth Anárion son of Elendil was slain, and many others. But at the last the siege was so strait that Sauron himself came forth; and he wrestled with Gil-galad and Elendil, and they both were slain, and the sword of Elendil broke under him as he fell. But Sauron also was thrown down, and with the hilt-shard of Narsil Isildur cut the Ruling Ring from the hand of Sauron and took it for his own. Then Sauron was for that time vanquished, and he forsook his body, and his spirit fled far away and hid in waste places; and he took no visible shape again for many long years.
I can answer your question at least partially, but I don't think the complete sequence of events is ever described in detail in book-canon.
The first time we hear about what happened (In publication order) is in Fellowship, during the Council of Elrond. Elrond, who participated in the battle and saw a lot of this first-hand, recounts it like this:
Gil-Galad died, and Elendil fell, and Narsil broke beneath him; but Sauron himself was overthrown, and Isildur cut the Ring from his hand with the hilt-shard of his father's sword.
So we know that Sauron survived Gil-Galad and Elendil. What's not clear, however, is who exactly killed Sauron. If we take Elrond's statement to be a literal sequence, then I would guess that Elendil struck Sauron down but died in the attempt, and all Isildur had to do was cut off the ring to render him completely powerless.
This is (in)famously not the interpretation favoured by Peter Jackson, who in the prologue to the Fellowship movie has Sauron kill Elendil handily, and then Isildur gets lucky and cuts off the Ring, defeating Sauron. Personally this is the interpretation I favour as well, because it's more consistent with how much everyone fears Sauron getting the Ring back; they don't treat that eventuality as "We could stop him again, it would just suck and kill a load of people", they treat it as "Game Over".
Unfortunately I don't have access to my copy of The Silmarillion at the moment, so I can't directly address how that one reads.
There is no ambiguity. Tolkien was not one to be sloppy with words. "Thrown down" and "overthrown" are synonyms for each other. They are not synonyms for "killed." Elendil and Gilgalad were the ones who "threw down"/"overthrew" Sauron. He was beaten, defeated, knocked to the ground and -- temporarily -- unable to get up or defend himself. He was not dead, though, and the Ring would have healed him. But he was vulnerable. Even in that state, Sauron was still terrifying, and there was no way to know how long Sauron would remain vulnerable, so let's give Isildur his due. Isildur seized the opportunity. He had the courage to approach the monster, and the wits to realize that cutting off the Ring would be Sauron's deathblow.
But of course, it wasn't really Sauron's deathblow, as long as the Ring still existed. Isildur had to have known that what Elrond said was true. That's what makes Isildur's fall so tragic. He basically said, "Sauron won't recover from this until I'm looong gone, so what do I care? I'm keeping the Ring to help me in the here and now." (That was his attitude. It's not a quote.)
So, to answer the OP's question: Sauron's body was destroyed, in the sense of being broken and "cast down," by Gilgalad and Elendil BEFORE the Ring was taken from Sauron. HOWEVER, it was the removal of the Ring by Isildur that prevented Sauron from using the Ring to heal his broken body, causing Sauron to "forsake" the useless body. I like to think that it fell to ash, like in Jackson's movie, but that's not canon.
A new answer to a very old question, but I stumbled on some relevant material that does not appear to be mentioned elsewhere...
Sauron's body was still in existence after Isildur took the Ring. Of the Rings of Power and the Third Age says
Then [after Isildur took the Ring] Sauron was for that time vanquished, and he forsook his body...
The related question as to who 'killed' Sauron (whatever that actually means for a Maia) is answered in letter 131:
Gilgalad and Elendil are slain in the act of slaying Sauron.
This is in agreement with the account in Of The Rings of Power and the Third Age, and the briefer account given at the Council of Elrond, in which Sauron is 'thrown down' (Silmarillion) or 'overthrown' (LotR) by Elendil and Gil-Galad before Isildur takes the ring. Isildur's later remark "Was it not I that dealt the Enemy his death-blow?" can be taken as hubris: the Ring had already started its work. Alternatively (as suggested by @Wad Cheber) it might be that Sauron would recover (rather than leave his body) had the ring not been taken. However this later event is interpreted, letter 131 is definitive: Gil-Galad and Elendil killed Sauron.
Firstly, Sauron did not die, ever. He is neither an elf nor a mortal, rather he is a Maya, sort of the equivalent of a Greek lesser God. Even when Frodo finally destroyed the Ring, Sauron still lived.
Sauron has however seemed to perish before by going to the Abyss when Numenor fell and then coming back, though less powerful than before. As Gandalf described it (paraphrasing), once the Ring were destroyed, Sauron would be reduced to a shadow gnawing at itself in the dark. Dead to this world for all intents and purposes, but not technically.
So if it is a question of what happens to Sauron's physical form once he wrestled with Gil-Galad and Elendil and had his finger cut by Isildur, there is nothing that Tolkien wrote that goes to specifics as to whether his body remained intact, conscious, turned to ash or what.
One thing that is stated in the FOTR is that Gollum after having been captured and personally tortured by Sauron described him as having nine fingers. This suggests that the physical act of Isildur cutting Sauron's finger had a lasting effect. So either Sauron's body remained somewhat intact (conscious or otherwise) or was disintegrated and then reconstituted with a missing finger.
As Gandalf is also a Maya, (but of a lesser order to Sauron according to his own words), Gandalf had fallen in battle against the Balrog, but was somehow brought back to life. Again it's not made clear whether Gandalf's body maintained a physical presence through the process, or if Gandalf was temporarily non-corporeal. I personally imagined the former and feel the same is true for Sauron.
