1.)  My suspicion is that the same folk who carved the Púkel-men that line the road also carved the signs on the door. 

2.)  Hmmm...  I suppose one could cheekily suggest that Tolkien had a "senior moment" and forgot that Legolas was not the only Elf in the party.  One might argue (as you do) that since Elladan and Elrohir had not yet made a choice as to whether to share the fate of Elves or of Men, that the same fear which affects other mortals applied to them as well.  Legolas, on the other hand, has no choice as to which kindred's fate will befall him: he is an Elf and thus bound to Arda no matter what happens.  The ghosts of men therefore hold no terror for him. 
      But I like Kimi's idea too. 

3.)  One possible answer (with wajeff) is that it is Gimli whom Tolkien imagines as ultimate narrator: remember, the whole story is told by Legolas and Gimli to Merry and Pip in "The Last Debate."  In the first version, the entire story of the journey on the Paths of the Dead is told in "The Last Debate."  Tolkien later decided to transfer that material to a much earlier chapter, but to retain the perspective of the original narrator.  It's one of the very few glimpses that we get into Gimli's mind, because here Gimli tells the story.

4.)  It's Baldor, son of Brego.  In the first draft (The War of the Ring = History of Middle-earth, vol. VIII, p. 407) Tolkien actually had Aragorn identifying the skeleton.  This was when all the material still stood in "The Last Debate" -- when Tolkien transferred the material to an earlier stage in the story, so that it stood just before Théoden's telling of the story of the ill-fated Baldor, he decided it was too much to have Aragorn identify the man as well.  All the same, Aragorn, who had spent a few years amongst the riders, indicates that he knows who it is: "Nine mounds and seven there are now green with grass, and through all the years..."  He refers to the royal tombs of the Rohirrim -- i.e. he knows that the skeleton belongs to that line. 
     Now, for me the mystery is this: what was behind that door?

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All that is gold does not glitter,
Not all those who wander are lost;
The old that is strong does not wither,
Deep roots are not reached by the frost.
From the ashes a fire shall be woken,
A light from the shadows shall spring;
Renewed shall be blade that was broken,
The crownless again shall be king.