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 may well have thought that this was how humans greeted each other, and was mimicking her facial expressions as well.
E.T. are the first and last letters of Elliot's name. Word of 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.
A cut scene restored to the 20th Anniversary Edition confirms this somewhat, as it shows Elliot introducing him to the bath, and E.T. climbing into the water and thoroughly enjoying immersing himself. Elliot is initially worried that his new friend might drown, but then realizes that this is what E.T. considers to be having a good time.
Some of the stuff All There in the Manual you can figure out just by paying close attention to the movie. That E.T. and his people are from a lower-gravity planet is fairly obvious from how they waddle awkwardly around on ours, and from how E.T. himself rarely ever stands anywhere near to his full height; the gravity is obviously weighing hard on him. Also, while one might not be able to discern that he's part of a Hive Mind, his telepathic link to his people through those glowing hearts of theirs certainly hints they're all rather closely connected mentally and physically; and when Elliot insists that "We're fine" (instead of "He's fine") in response to Michael's concerns about E.T.'s health, Michael points out that he's finding Elliot's new manner of speaking ("You say 'we' all the time now!") a bit disquieting. Finally, the wilted flowers E.T. revived (which began drooping again as he got weaker) and the doctors' various vital-sign scanners make abundantly clear that he had the power to transfer life to and from others and was piggybacking on Elliot's vitality to survive until he could get back to drawing it from his own people.
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 Heartwarming Moments: 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.
Of course, all that stuff with the Jedi and Sith took place "a long time ago in a galaxy far, far away." There have to have been some major political changes over all those eons, and the books indicated getting to other galaxies even with those hyper-drives everybody had was a daunting proposition. One of the orchestral pieces' titles on the soundtrack also indicates that in this story, E.T. is 3 million light years from home. Who knows how long it took his ship to get here, how long it might take for it to get back, and who's running things over there these days?
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.
More an example of Fridge Squick, perhaps: E.T. has no obvious openings between his legs in front or in back, yet he's shown consuming food. Considering his physical frailties, I doubt he and his people are able to digest everything they eat into energy and nutrients, so they must have some waste products. Where does the stuff go, then? Most likely, out through their skin pores. That white stuff covering E.T. later in the film when he's sick? That's the waste his body is no longer operating efficiently enough to shed as quickly as it did when he was healthier. The scientists gathering stuff into containers were surely able to gather plenty of E.T.'s genetic material from all the dead skin cells he was shedding around Elliot's house for days along with his powdery white waste.