[Elladan and Elrohir]

It looked to me and my brother like there were two dead orcs moving, or maybe even three, but I'll have to watch the scene again before I can make a definitive statement.

I don't think I've ever heard of an airplane flying over during filming, but there is documentation of a car in a scene from FOTR. Here's the full story (from amazon.com):

On the film's first theatrical release, a story circulated (and was reported in the goofs section) that when Sam tells Frodo that he is now the farthest he has ever been from home, a car is visible driving by in the background (top-right corner of the screen). Arguments ensued. Some said it was smoke from a chimney, others said they saw the glint of sunlight reflected from the windscreen of a fast moving vehicle. In the version of the film released on DVD there is definitely *no* car, only chimney smoke and a one-frame flash of light which *could* conceivably be a car, but not in any sense that could be considered a goof. Jackson says (in the commentary track on the Extended DVD) that he looked at every frame on a computer and has never seen anything resembling a car and claims that it's nonsense (and certainly the original sighting remains unconfirmed by IMDb goof spotters). In the documentary of extended DVD version, John Gilbert, the editor, says that there was a car in the background, but they thought no-one would notice it. They got rid of it in the DVD version. In a subsequent magazine interview, Jackson agreed that the car was there.