Why didn't Elliot go on adventures in space with his best friend? The answer becomes clear throughout the movie. At the beginning, he's a very typical 10-year-old, hurting from a divorce, and unable to keep that hurt to himself, even making his mom cry. His brother yells at him for that, telling him to grow up and think how other people feel for a change. When he comes back from sleeping overnight in the forest, he learns that he is just as capable of hurting his family as his father did when said father left his three kids and wife, and at the end, cannot bring himself to hurt his broken family further by leaving with the aliens in the spaceship, no matter how much he loves one of them. Ouch.
Which makes it Spielberg's rebuttal to the ending of his previous Close Encounters of the Third Kind where the father did leave his family to go gallivanting with aliens (none of whom he had formed a relationship with).
When E.T. sees Gertie for the first time she screams and he screams back. He was scared of Elliot at first, screamed, but came back and got to know Elliot. By the time he met Gertie and she screamed, he thought that was how humans greeted each other and was mimicking her facial expressions.
E.T. are the first and last letters of Elliot's name. Wordof God, a trading card series to tie in with the movie says the family's last name is Taylor. Elliott could've given E.T. his initials.
Then, because Elliot relates to E.T as an outsider, the film could also be called Elliot Taylor (:The Extra-terrestrial part of the title hinting that Elliot also sees himself as an alien due to him being different and alienated after his father left home).
"E.T." being short for "extraterrestrial" carries a double meaning too. "Extraterrestrial" (one word) conveys the obvious meaning that he's from outer space, sure. But if heard as "extra terrestrial" (two words), it means very terrestrial, i.e. Closer to Earth. E.T. is a botanist who knows about nature, and he comes literally "closer to Earth", both physically (he's stranded there) and emotionally (he learns and cares about an Earth family), than any other of his kind.
Why does Michael find E.T. by a river? Word of God states that E.T. is a plant like creature (The Expanded Universe book and games touch upon this) and he could've been there to help himself survive.
Why the FBI are keeping their distance during the chase could mean two things, and there's a subtle payoff to how careful they're being. One: They are following the kids to try to find where the spaceship is. But Keys is ultimately the only person outside the Taylor family to get near the ship. It's probable they were letting everyone go in peace. There's also a good chance the rest of them got to see their takeoff. Two: while the kids are harboring an alien, they are still kids and they are taking extra special care of not speeding up in risk of running them over. Even with the guns pointing at them, they were more or less a last resort warning, or else they would have shot them down during the flight. The guns probably weren't even loaded at all. They just wanted to intimidate the kids. The agents obviously don't mean any harm throughout the movie, they're just trying (and ultimately failing) to help and reach out to higher intelligence.
Also counts as Crowning Moment of Heartwarming: Before Gertie meets E.T, she yells "Elliott, look what I made for you" and can be seen holding a piece of paper in her hand. Elliott faked being sick so he could stay home and play with E.T, and Gertie genuinely thought he was sick and made a get well card.
The movie is basically a retelling of The Wizard of Oz. E.T. wants to get home and is helped by three companions. Some scenes are even similar, such as E.T.'s tearful goodbye to each of his friends at the end.
E.T. is able to make the bikes fly, as well as other objects. It's possible he can't make himself fly. At the beginning of the film, why didn't he sit on a tree branch or rock and fly back to the ship?
He would have to stop and find a rock that would support his weight. He can't just grab any rock he finds, and it would need to be light enough to levitate. Levitating a branch would likely require removing the branch from the tree, which is unfeasible.
A minor example. Since we don't know what happened after E.T. left, it is possible Elliott's family got in trouble with the federal government for setting E.T. free.
A CrackedAfter Hours video argued that if E.T.'s mind-controlling, telepathic, lifeforce-draining, telekinetic, resource-harvesting race were ever to decide our planet was worth conquering, we would be in for a Cosmic Horror Story. Luckily, E.T.'s species seems to be as peaceful and sweet as he is.