J.K.R. Doesn't tell us, but I think that she should write another book about his kids and one of them finds it.
Likely not since he dropped the resurrection stone in the forest and never returned to look for it, believing it to be safer where it was.
Albus Dumbledore found the resurrection stone and hid it inside the golden snitch which he left to Harry in his will. When Harry told the snitch he was about to die, it revealed the snitch to him. Harry dropped it in the forest and never went back to look for it, deciding it was better for the stone to be lost forever.
the forest again
Marvolo Gaunt did, he hid it inside a ring which Voldemort made into a Horcrux. When Dumbledore destroyed the Horcrux he found the ring and hid it inside the first snitch Harry ever caught. Harry dropped it in the Forbidden Forest and never went back for it.
It died when voldomort killed Harry in the for ridden forest
In Harry Potter, once you have died you cannot be brought back to life. Sirius did reappear when Harry used the Resurrection Stone but only as a spirit who disappeared when Harry dropped the Stone.
Harry kept the Invisibility Cloak so that he could pass it down to the next in line. He left the Resurrection Stone in the forest clearing where he had dropped it before his "death" and never went back to look for it, nor did he tell anyone else where he put it. He says in the final chapter of DH that he would be returning the Elder Wand back where it should be, which, I presume, is Dumbledore's grave, so that, if Harry died a natural death, its power would be broken. Hope this helps.
The resurrection stone was actually the ring Dumbledore found between the storylines of The Order of the Phoenix and The Half-Blood Prince (The ring that ruined his hand, since it was a horcrux)
Yes. In the chapter 'The Forest Again' Harry opens the very first snitch he ever caught and the resurrection stone was inside it. When Harry held if he saw his parents, Remus and Sirius again (though they were not alive). After he drops it he never sees him again, except in pictures.
The Resurrection Stone was given to Harry by Dumbledore, who had hidden it within the first Snitch Harry caught during a Quidditch Match.
he ran away and jumped in the tree with harry
The resurrection stone was originally owned by Cadmus Perverell. There are many gaps between generations but at some point it fell into the hands of Marvolo Gaunt who gave it to his son Morfin Gaunt. Then Tom Riddle (Lord Voldermort) killed his uncle Morfin and stole his ring thus making it into a horcrux. However he left it in the same house and because Dumbledore knew his true origins he found it. He then left it to Harry Potter inside the very first snitch Harry ever caught (which he caught in his mouth). Harry dropped the stone in the middle of the forest and he never went looking for it.