Dory's Childhood Friend, Destiny, is a whale shark, which is how she learned to speak whale (they were "pipe pals").
Continuing from above, it's shown that Destiny can both speak whale and speak normally in the movie, and that she prefers to speak normally rather than whale, because she's technically not a whale, but a shark. (Genuine whales like the one in the previous film can only speak whale) It's possible that Destiny herself also learned to speak whale because she lived next to Bailey's, a beluga whale, exhibit, and that Dory herself subsequently learned it from her.
Alternately, whale sharks may be bilingual in Whale and - er, whatever the other fish speak - because they talk to baleen whales all the time about where the richest plankton can be found.
Another alternative: Whale just carries better through the pipes.
Destiny's also the first shark Dory ever knew, which helps explain her placid acceptance of the "Fish are friends" sharks in the first movie.
It seems Marlin treats Dory like she's incapable with her short-term memory loss, and that's understandable. But shouldn't he have learned that people with shortcomings can accomplish anything, especially after his adventures in Finding Nemo? Actually, no. Because in "Finding Nemo", Marlin only learned how people with physical handicaps (Nemo's 'lucky' fin) are capable. "Finding Dory" is about Marlin learning that people with mental handicaps (Dory's short-term memory loss) are capable as well.
Dory's ability to read is never explained in Finding Nemo, but it is here: she was born in an aquarium and surrounded by exhibits.
In The Stinger, we see the Tank Gang from the previous film, still trapped in plastic bags like before. After all the time passed between then and now, the bags are now filthy, except for the one containing Jacques the shrimp. Why? Being the cleaning shrimp, he cleaned his bag himself.
Combine this with Nightmare Fuel: Did Nemo and Nigel forget about their friends? They'll die in there for sure!
Nemo is at least excusable, as he didn't know they had escaped. Nigel, however, has no such excuse considering the long-standing relationship he had with the Tank Group.
They're being taken to the Marine Life Facility, where it is implied that they will be treated and taken care of until they are strong enough to be released back into the ocean, so there's not much to worried about. This begs a Fridge Logic, though. How did they survive so long inside bags without food or a renewable source of oxygen?
When Dory relives the memory from the night she's forcefully taken away from her parents, she's in the little alcove where she used to talk to Destiny. Because of this, she was able to overhear her mother worrying for her, and so she went off to find a purple shell near the pipes, where she was whisked away into the ocean. In a way, Destiny was what brought Dory, Marlin, and Nemo together.
Hank despises being touched due to a traumatic injury in his past, but over the course of the movie seems to grow accustomed to Dory's touch to the point where she can outright hug him and his only reaction is to look surprised. It's a nice, subtle way to show that he's slowly coming to accept her as a friend and it makes his decision to go back to the ocean with her instead of Cleveland more understandable.
The fact that Dory's opinion on sand is the exact opposite of Anakin Skywalker's has not passed unnoticed, but if you think about it, it's not really a difference of opinion. Dory describes sand as "squishy" while Anakin describes it as "coarse, rough, and irritating"—but considering one lives underwater and the other grew up in a desert, those aren't their opinions, those are facts. No wonder one finds sand pleasant and the other doesn't.
It's also a Shout-Out to Dory naming the baby jellyfish 'Squishy' in the first film.
In addition to that, she managed to find someone who was looking for their family. Something in her knew that's what she was looking for. She might not have been able to find hers, but she helped Marlin find his. And because of that, she found hers, too.
In what may well be both a Call-Back and a Call-Forward to Finding Nemo, remember how Marlin got Dory through the jellyfish despite her poor memory and easily-distracted nature? He framed it as a game! Now we see Dory's childhood and learn that her parents did the same thing, trying to impress important lessons on her (how to introduce herself, how to find her way home) in the form of games (hide-and-seek, collecting shells).
Why did Hank cheat and try to steal Dory's tag to get to Cleveland? Because he knows his tentacle would eventually grow back, which means he will be considered cured and released, which he doesn't want due to his trauma.
Cleveland is where fish in Quarantine go if they can't rehabilitated and released. Dory was in pretty good shape when they brought her in though, so why would they tag her for Cleveland? Because they were sending the blue tangs to Cleveland already. Sort of makes her adventure through the aquarium a bit of a "Shaggy Dog" Story, since she could have just stayed in Quarantine and found the other blue tangs that way.
Except, if she had stayed in Quarantine, she'd never have met Destiny and Bailey, so wouldn't have been able to turn to them for help later.
She wouldn't have been able to find her parents in Cleveland, so staying in Quarantine wouldn't have helped.
Which is still a kind of a different "Shaggy Dog" Story as Dory didn't need to go into the Marine Life Facility in the first place, as her parents have escaped out and made a small enclave somewhere in the kelp forest where she, Marlin and Nemo first arrived. If they had searched around the kelp forest first instead of going to the surface, they would've eventually found Dory's parents and Dory wouldn't have got caught by humans and sent to inside the facility. Of course, she wouldn't have met Hank, Destiny and Bailey, and the marine animals in the truck would've been sent to Cleveland, so...there's that.
Marlin was a bit crankier than usual for much of the film. Why? Because he had to get up early to find Dory after she sleep-swam and lack of sleep can affect one's mood.
Her sleep-swimming was a missing trailer scene, but she did end up waking up Marlin and Nemo rather early so the brilliance still applies.
Bivalves can evidently speak, as evinced by the talking giant clam. So why didn't Dory ask any of the mussels inside the pipe network for directions? Probably because, although they might've been able to talk, they're immobile and blind, so wouldn't have had a clue which way she needed to go.
Dory's home, mere inches away from Marlin and Nemo's anemone, is brain coral. Brain coral with a hole in it.
Why is Becky so ditzy and unable to talk? She's a loon, and the term "loon" is frequently associated with insanity. Becky is the way she is because she's crazy!
When Marlin and Nemo are trying to get out of a gift shop tank, they do so by thinking "What would Dory do?", but Marlin's initial answer to that is more along the lines of "What would Marlin do?". During Dory's Heroic BSoD, she asks herself "What would Dory do?", and proceeds to analyze her surroundings, coming to the conclusion that being in the kelp forest would be better than the open water, and navigation would be easier from the seafloor; logical deductions that Marlin would be more likely to make, and they ultimately lead her to her parents. Basically, Marlin and Nemo got back to Dory by answering "What would Dory do?" and Dory got back to her parents by indirectly answering "What would Marlin do?"
Destiny recognizes adult Dory when she says "You're welcome!" in whale. Later, a flashback shows Baby Dory speaking to Destiny through the pipes and what is she saying? "You're welcome!" Destiny didn't recognize her just by her accent, but by that specific phrase.
The post-credits scene shows The Tank Gang being picked up by humans to be rehabilitated at the Marine Life Facility and released back into the wild. Keep in mind that one member of the group is Bloat who is a porcupine puffer fish. For those not familiar with puffer biology, their teeth constantly grow throughout their lives. They need to eat hard-shelled foods (such as crabs, shrimp, or clams, for example) to wear down their teeth. Otherwise, they risk their teeth overgrowing which can cause the fish to starve. Even worse, overgrown teeth can puncture through the skull and cause serious injury or even death. Now keep in mind that Bloat has been trapped in a bag without anything to wear down his teeth for months. It's a good thing he was somehow lucky enough to accidentally end up in a place where humans can take him in, give him the proper diet, and trim his teeth down to a more comfortable size. Otherwise, he'd end up dying a slow and very agonizing death. Suddenly that little bonus scene doesn't seem so funny, huh?
Bloat needing to have is teeth filed would be the least of their worries since they somehow managed to survive for a year inside plastic bags without apparent food or new oxygen. From the Tank Gang's perspective, the fact that they arrived at a Marine Life Facility means that they're pretty much back at square one. "Fish aren't meant to live in a box".
"Rescue, Rehabilitation, Release." They're gonna be fine.
Well, unless they're so sick that they get tagged for Cleveland... that would be even worse than square one.
Although presumably at a professional aquarium they'd have plenty of room and proper care (i.e., the Open Ocean exhibit), not a small amateur fishtank and a visiting Darla.
Hank lost a limb due to some sort of unexplained incident. Octopuses can normally regenerate lost limbs. However, for some reason, Hank is unable to regenerate his limb. Whatever happened, it was not only traumatic enough to give him PTSD, but severe enough that it caused him to be unable to regenerate.
It's implied he received the injury in the wild, causing him to be brought into the MLI as a rescue. His traumatic experience is presumably the source of his desperation to be moved to the Cleveland aquarium before his arm regrows, he's considered rehabilitated, and MLI releases him back to the ocean.
It might be due to mating. Male octopuses have one tentacle that is used to deposite sperm into females, and females are much larger and will sometimes eat their mate. To avoid being eaten (which in and of itself is a very traumatizing experience), he might have detatched his tentacle in order to survive.
Had Dory not been separated from her family, while it would've been positive for Dory, imagine the implications it could've had for Marlin, assuming he was unable to find help from someone else, its quite possible that if Dory had stayed with her family, Marlin may had never found Nemo, as Dory being capable of reading was able to discover that Nemo was in Sydney, she was able to convince the school of moonfish to lead them to Sydney as they refused to help Marlin initially and without her the Whale would've not helped Marlin reach Sydney.
While it's seen as a good thing that the fish being shipped off to Cleveland were released into the sea when the truck went off the highway, the fridge horror comes when you realize that the fish destined for Cleveland were not supposed to be released into the wild. Of course, what happens to the fish in Cleveland or why they're shipped there wasn't revealed.
They were specifically destined for a Cleveland aquarium, so presumably they would be living quite well. (Dory's parents were understandably happy being part of the MLI's permanent exhibit until they lost their daughter.)
This may have been karmic, since the MLI's claim of rehabilitation and release doesn't seem to be true. There was nothing wrong with Dory, yet she was tagged for Cleveland simply by virtue of being a blue tang. Even if the humans could tell she had memory issues, she was tagged way too soon for them to have noticed. Also a number of exhibits that were there just for show, like the touch pool. And what was wrong with that clam?! Most of the animals seen had nothing apparently wrong with them. The MLI seems more like Sea World in that it claims to have pure motives, but is more interested in money.
Let's make the point that they actually do rehabilitate and release animals; look at Fluke and Rudder. Dory was tagged for Cleveland because she's not native to Californian waters. They probably thought she was some exotic fish an irresponsible pet owner had dumped in the sea; a fish that could not survive in Californian waters. The whole thing smacks faintly of a last-minute script revision (apparently, a Sea World-esque marine park was going to feature in the script; it got changed at the last minute after Blackfish; quite why, considering the ending involved animals escaping back into the ocean is beyond me).
The Marine Life Institute is probably going to have a hard time dealing with the aftermath of the events of the film: Three marine animals have escaped from captivity, two of which are their star attractions in show exhibits (Bailey and Destiny) and a truckload of marine animals that, as mentioned above, are not supposed to be released into the ocean, is hijacked and crashed into the sea in a 'bizarre accident', releasing all the contained animals within it. Somebody is going to be so fired from this.
Not to mention the massive traffic accidents Dory and Hank caused. At the very least there are serious injuries ...
While it is sweet that Dory's parents went out every day lining up shells to guide her home, it's also pretty creepy. They've done this every single day for years now. What if Dory had never remembered and went looking for them? They'd have spent every day wondering what happened to her, never being able to live their own lives ...
We see a flashback of little Dory in the same scrapyard, where they encounter the squid. She was calling out loudly just like her adult counterpart, and kept getting shushed by the denizens of that area. What are the chances that young Dory encountered the same squid years before and barely escaped with her life?
Dory's friends might find themselves uncomfortable with the three sharks occasionaly hanging around the reef. Oh, wait, the sharks are nice and don't eat fish. Then you realize that Hank and Bailey are not fish.