You buy them with 8 plat from a vendor (named Jerita?) located at Ydra Forest - Traders Outpost
I'm guessing once you buy a jewel elixer, it instantly starts your 15 minute timer (I haven't bought one to test yet)
Chance of looting a specific jewel of the elixer type is based on damage dealt (so higher damage dealt = greater loot chance (i think...))