After reading all this, my opinion is that the fire elemental would not sink, but would indeed float, or, as was so eloquently put above, skitter about on a cushion of steam. I'd also have the character be completely freaked out...
ALso of note: I'd have to rule that the fire elemental is not harmed by being in water... although, as I stated above, also cannot BE in water. Fire is not actually extinguished by water. Not at all. In fact, you can burn several substances under water. People weld underwater with acetylene torches. There is fire. What happens is that water typically seperates the fire from it's source, and not having a source extingushes the fire.
Now, assume that we do have the character go underwater, say dragged there by weight or something... OR perhaps teleported there... Do the rules say they need to breath? Or that they need oxygen to burn??
I'd likely treat them as any other character who does need to breath going underwater, with the exception that I wouldn't allow the character to attempt to swim. Run, perhaps. I see that freaking out and running around as a very valid option.
I can't see the character taking any damage whatsoever from the water. Although I can see the character taking damage from the cold if the water was cold...
Actually, after some thought I'm most liklely rule that the character could not run, or really move, and would float to the top after getting rid of enough weight. At which point it would skitter around. I would allow the character enough movement to drop things and remove armor.
Of course, I'd have some issues with exactly what armor a fire elemental would be allowed to wear...