Dáin set out with an army of Dwarves to the aid of his royal cousin. It is uncertain whether Dáin's son Thorin accompanied that army (he was certainly old enough to do so by that time, but he may have remained behind to ensure the survival of the line of lords; at the very least, no mention is made of him in the annals of those events). At Erebor, Dáin fought in the Battle of Five Armies against an unforeseen incursion of Orcs, and though the Orcs were defeated, Thorin II was slain in that battle, as were both the remaining heirs of his line, Fíli and Kíli.
Thorin's epithet of Stonehelm is as mysterious as almost everything else about his life, and we have no account to explain how he received it. It was most likely acquired in battle, perhaps at the Battle of Five Armies or the Battle of Dale, though the detailed circumstances of the title's origin are not known.
We have no explicit record of the date of Thorin's death, but the Dwarves of Durin's line typically lived to the age of two hundred and fifty, with very little variation from one generation to the next. It's safe to assume, therefore, that Thorin III lived until about the year IV 96.
It seems likely that Thorin III received his name in memory of his illustrious forebears Thorin I and (especially) Thorin II Oakenshield, who had recently become the King of Durin's Folk at the time of Thorin III's birth.