Just an FYI regarding the opening piece - Hillary indeed wore a Rolex Oyster Perpetual COSC. What he wore on the expedition was not an Explorer, because that watch didn't exist yet. He wore an Oyster Perpetual COSC with a white dial on a leather strap. He actually wore two watches, one on each wrist, and one of them was the Rolex. The other was the Smith.
To celebrate the achievement, Rolex designed the first Explorer, based on Hillary's Oyster Perpetual, to commemorate the watch.
Hillary confirmed this in letters, as well as photographs from the expedition, and his actual watch is now on display at the Beyer Clock & Watch Museum in Zurich so this has all been substantiated.
Here's a great link that goes over all of the information, tells the story, shares the letters from Hillary, the photographs, etc.
http://rolexblog.blogspot.com/search/label/Mount%20Everest
Here are some highlight pics:
This is the actual watch worn on the expedition.
Rolex ad containing Hillary letters.
Photos of the watch itself.
Hillary wearing the watch. Notice he is wearing the two watches, one on each wrist, in the group photo taken just after they descended from the summit.