It seems we do, for completing quests. What about killing monsters?
Patch 1.0.4 introduced a new level system for level 60s called Paragon Levels. There are 100 Paragon Levels. Killing monsters and completing Combat Challenge Streaks (Killing Blows, Massacres, etc) will award EXP towards Paragon Levels. Leveling up will give you more core stats (just like regular leveling up) along with +3% MF and +3% Gold Find per Paragon Level. You also get a new border around your char that displays your Paragon Level to brag to your friends.
When you gain level 60, all exp is gone. Killing things will no longer give you XP, items and gold. It will just give you items and gold as usual. Quests will only give you gold instead of XP and gold. The nicest thing about level 60 is the Nephalem Valor which increases your gold and magic find at level 60 every time an elite mob is killed. Each buff gives you 15% more gold find and magic find so it stacks to a max of 75. This stacks up to 5 times and will last 30 minutes from your last elite kill, assuming you do not change any skills on your character and remain in the same game and act.
So in short no, XP just disappears and gold remains the same as the other levels. Think of it this way, All XP gained at level 60 is now 0. Nothing else changes EXP wise.
From my experience, it seems like all the XP rewards are just not there - you don't get additional gold from killing monsters, earning bonuses (destruction, massacre, etc) or completing quests, the XP rewards are just gone.
You do gain access to the Nephalem Valor buff, which increases your gold and magic find as you kill champion and elite monster packs, however.
No, The gold will drop as it normally does.
You just don't get any exp for killing monsters the drops are still the same.
But You'll still get gold from quests.
If you find those log books and such you notice there is a +exp that pops up when you grab it but it is a lie and you get no exp from it. You just get to hear the beautiful recording.