In this picture it looks like they used the mantle and didn't hang it from the rock. If you can do this, even better. :)

In order to hang from the rock, you would need to utilize wall anchors especially made for stone. Usually you can get these at your local hardware store. Ask their advice as to what anchors to use. It is easier to drill into the mortar between the rocks than the actual rock.

Depends on the stone.. softer sandstone/limestone can be safely drilled with a rotary hammer and carbide masonary bit.. and then inserting an anchor, as mentioned above (and also as mentioned, any mortar will be easier, but picture centering may be harder) or use special purpose masonry screw (EG Tapcon or Spax). Harder stone esp granite will benefit greatly from a diamond dust tipped drilling bit.. This bit DOESN'T like hammering from a rotary-hammer drill) , but DOES like water spray lubrication and slow drilling speeds and pressures.