How did 10 yr old Elliott [not even a genius as portrayed in the film] develop the makeshift communicator [used by E.T. to phone home] which scientists from all over the world have failed to develop till now?
In the book, E.T. was in mental contact with Elliot the entire time and guided him. E.T. told him which parts he would need and how to construct them to be able to send a message to the mothership. I think it's important to note that scientists have, in fact, developed communication technology that's superior to what we see in the film: The space ship was in orbit waiting for a signal from E.T. to let it know when he was ready to be retrieved, E.T. didn't actually phone all the way home. This is also supported canonically in the film, as @kevinhowell suggests in the comments, by a scene where E.T. drinks and Elliot also acts as if he's drunk in the middle of school.
As an aside, E.T. was sure that the youth he was dealing with were a seperate species of bipedal ape, and that they were far superior intellectually to the large, slow, stupid ones. He was also very impressed with the holiday "Hollow Bean", during which the small creatures dressed up and stole sweet treats from the large ones. (This additional information comes my memory of reading The Book of the Green Planet.)