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Azeroth is a planet which the Titans used to contain the Old Gods. There may be other Old Gods imprisoned in other planets, we don't know. Draenor may have housed an Old God or two prior to it's explosion.
Neltharion did indeed explore the depths of Azeroth, and in his travels the Old Gods did hear the whisperings of the Old Gods. It is entirely likely that he heard all their whisperings, not just one or two.
The Old Gods encouraged Neltharion to create the Dragon Soul/Demon Soul artifact during the War of the Ancients, and infused it with some of their power, along with that of the Aspects. When Neltharion attempted to use it, it literally tore his body apart. He then retreated to his underground caverns, and had the plates created and fitted to his body in order to hold it together. He then took on the name Deathwing.
The Deathwing we see at in the Well of Eternity dungeon has already had this event take place, already betrayed the Aspects, already wiped out the majority of the Blue Dragon Flight (including Sindragosa, the mate of Malygos who you may recall from ICC), and already had the plates bolted to his body.
Side Note:
Meanwhile, Illidan recovers the Dragon Soul and hands it over to Azshara to help her open the portal to Sargaras. The fact that we see a Demon Tattooed Illidan in the Well of Eternity dungeon means that he had already betrayed his people and been marked by Sargaras himself. So while he is in fact trying to help shut down the portal at this point, he has already betrayed his people, and been rewarded with demonic power.
In the dungeon event, he tries to take back the Dragon Soul, and the Old Gods prevent him from doing so, and further warp his body.
In the years between the Well of Eternity and now, The Dragon Soul falls into the hands of the Orcs when they arrive on Azeroth. They use it to subjugate the Red Flight and use them as mounts in their campaign in Warcraft 2.
Deathwing also has a hand in the Elemental War.
Deathwing's ultimate goal is to bring about the Hour of Twilight using the Twilight Dragon Flight, the ones Sinestra (Deathwing's Mate) was working on. You actually get to see the result of this in a dungeon called The End Times which is a possible future where the Hour of Twilight takes place. In other words, if we fail to kill Deathwing, this horrible end comes to pass.
They also wrap up the Infinite Dragonflight storyline in the End Times dungeon with the death of Murozond, the corrupted and anguished form of Nozdormu.
The Hour of Twilight basically ends all life on Azeroth, the Old Gods get free, etc etc.
Then you go into Dragon Soul, you fight the following:
-Morchok, the oldest earth elemental alive, powerful servant of Deathwing and the Old Gods
-Two Generals of the Old Gods, Vezax and Yorsahj
-A powerful Shaman/Twilight Cultist, Hagara
-Ultraxion, the ultimate Twilight Dragon, who's big bad attack is literally supposed to end the world, called Hour of Twilight.
-Thrall shoots Deathwing with the Dragon Soul. You then board an airship and attempt to catch up with Deathwing, only to be blocked by the final charge of the Twilight Dragon Flight, and a powerful dragon rider named Blackhorn riding his mount Goriana.
-You then parachute onto Deathwing's back, and due to Thrall wounding Deathwing, start tearing off his back plates to give Thrall a better shot. Sure enough, Thrall hits the now highly vulnerable weak point, and Deathwing plummets into the Maelstrom to his supposed Death.
-Deathwing rises from the Maelstrom, no longer contained by his Elementium armor, his corruption now at it's peak, and essentially turns into a giant Deathwing/Old God hybrid of some sort. With the assistance of the 4 remaining Aspects and Thrall, you finally defeat Deathwing once and for all.
Aftermath:
By expending so much energy on empowering the Dragon Soul to use it against Deathwing, the 4 Aspects have lost much of their power. They are no longer Immortal.
However, it should be noted that after the events of Firelands, the World Tree Nordrassil was again empowered by the Aspects as well, so the Night Elves have their Immortality back (in theory, not yet confirmed).
As of the novel involving the Emerald Nightmare, Teldrassil was similarly blessed. And the new Lich King, Bolivar Fordragon, who is also empowered by Alextrasza's Clensing Fire, make up 3 theorized 'containers' of their power.
The Age of Dragon's is over ending a period of Azerothian history spanning some 10-20 millenia, maybe much longer.
As for Old Gods, we know that Nzoth sleeps in the ocean and is likely responsible for warping the Highborn into Naga, and is probably good friends with Azshara. An Old God empowered Naga Queen with boatloads of Arcane power and was of considerable talent in using it before the Well of Eternity, and an entire race Naga at her command. Yeah.