Oh cool, a question I can answer...I think. The reason it was written in elvish was because the door/gate to Moria was built so that the Elves that lived in Hollin could visit the dwarves that lived inside the mountain. Hollin was the area surrounding that side of the mountain where the gate was. The dwarves, if they wanted to visit the elves, would just go out the door--it opened outward without any password or anything. And the elves were the only ones who would be passing into Moria from the other side, so they would be the only ones speaking a password to get inside that way.
Does that make sense?
(Jen, please correct me and fill in any relevant data, okay?)
Penny, the door of Moria is written with Elf-runes because Dwarves use Elf-runes and Elvish. Dwarvish is a secret language and they do not seem to have any runes of their own; they like Daeron's Runes (Cirth) better than Feanor's (Tengwar).
OT, as for the 'fallen' debate, the Elves are not perfect. No one says they are. They disobey the Valar and certainly sin, so it would seem they too are fallen. The Fall is not depicted in ME, but it seems a fact. As to what fallen means, according to the catechism I learned as a child, it means that everyone is "corrupt in every part of his being". Goes for Elves as well as Men, I suppose.