[Voralphion]

The doors were written in elvish because the western door was the one mostly used by the elves of Hollin when the dwarves and elves were friendly. Also, I think most of the time, the door was open, as a point is made when the war of the elves and Sauron is described in I think the Silmarillion, it says specifically that the doors of Moria were shut, (I don't know the exact quote). This could mean that they were normally open, obviously being guarded though.
As to why the name Moria was on the door since it was a derogatory term meaning black pit which it wasn't during its height, Gandalf could have paraphrased it, using the more common term at the time, Moria instead of Dwarrowdelf. I don't think it was a joke by Celebrimbor, as I'm sure the dwarves could speak and read elvish, as they are said to be good with languages, and since they didn't teach Khuzdul to non-dwarves, they would have to have learned elvish in order to communicate.