Yes, there's often some ammo or caps in them.
There's a shovel in Goodsprings near the water Source, or merchants will sometimes sell them. Chet always has one I think.
Yes, in my experience since release of F:NV, a shovel is ALWAYS in his inventory.
Also, if you're lucky, sometimes you find merc suits, most often I just find wastelander clothes
You can buy a shovel from Chet in Goodsprings. You can dig up graves but won't get labeled a grave digger like in FO2. No one seems to mind you exhuming their friends and families remains and picking through them.
i never found anything good in a grave except chance's grave.