The Arkenstone

[Gorhilion]

What happened to the Arkenstone after it was found?

[A_Brandybuck]

The Dwarves worked with the Arkenstone, so that the Stone became a very beautiful jewel. And it became a heirloom of the Dwarven-King of Durin's House until Smaug sacked Erebor.
Thórin and fellows got the stone back from Smaug and it was buried with Thórin under the mountains. That was it.

[Gorhilion]

Hmmm. It seems kind of a waste of such a beautiful jewel, to be buried with a dead person. It should be passed on to the generations to come. At least thats what I think.

[JennyHallu]

But this way it cannot be used against the dwarves as Bilbo used it to force an alliance.

[Gorhilion]

What happened to the rest of the treasure that Smaug hoarded after he was defeated.

[AragornII]

Smaug's treasure was distributed among the people living around the Lonley Mountain. Bard got a share to help rebuild Dale and Esgaroth (Laketown), the Elven King got some of it, Bilbo got his fourteenth share, and the rest just stayed with Dain, King Under the Mountain.

As for the Arkenstone, in the chapter "The Return Journey", it says

They buried Thorin deep beneath the Mountain, and Bard laid the Arkenstone upon his breast... Upon his tomb, the Elvenking then laid Orcrist, the elvish sword that had been taken from Thorin in captivity. It is said in songs that it gleamed ever in the dark if foes approached, and the fortress of the dwarves could not be taken by surprise.

This sounds like the dwarves had access to Thorin's tomb, if they could tell that Orcrist was glowing. So, this would mean that they could get to the Arkenstone too, unless they just had a little window at the door. I guess they just considered it too sacred to disturb. I think it could still be used like Bilbo, only it's much harder to get ahold of now. It could still be done though.