Is this just for story purposes, or is there a specific reason R2D2 can't speak English? I mean, obviously he can play back recorded messages, so it would seem he should be able to speak English if he wants to. Does he just...never want to? And how can human characters understand his beeps and boops?
I think it's just the way all astromechs are programed. It's not unique to R2, none of the astromechs we've seen have been able to speak English. The biggest examples of this besides are R2 are Chopper in Rebels, BB-8, and Bucket from Resistance. As for the other characters understanding him, I just assumed that his sounds are a language like any other and the other characters simply learned it. We do see Luke getting a translation when he's in his X-Wing in The Empire Strikes Back and Return of The Jedi, so that might be at least part of how he learned it.
I guess R2 wouldn't be as quirky if he could speak English. His beeps and boops are what make him funny...
I believe he actually did speak English in the first few versions of the script for the original movie. I read the comic based on Lucas's first draft script, and I'm pretty sure the version of R2 in that spoke English.
The mysteriousness behind the droids is something I have always enjoyed. I have wondered about them often.
I would love to get a novel or series going through the saga entirely from R2's perspective. Since 3P0 got his mind wiped, he's the only character who was involved in all 9 of the Skywalker Saga movies, plus at least one Rebels episode and a cameo in Rogue One, who actually remembers everything. From the way 3P0 talks to him, he seems to be a snarky little bastard, so a first person narrative from his perspective could be fun.
Nope. R2 always escaped memory wipes. He knew everything. The beeps and boops are called Binary in the setting, but conveying anything other than very basic info in binary would need much longer series of sounds.
Industrial Automation saved money by not building that capacity in, instead requiring an aftermarket sound-card for the droid to make noises other than basic beeps and boops. When you're dealing in bulk astromech purchases, those savings really add up.
If you have seen Solo, L3-37 has the brain of an R3 astromech, who wasn't able to speak Galactic Basic, but salvaged parts from other former droids and now can. But she also upgraded her mind, including memory and processors, so it's unclear which component gave her speech, or it is none, and she altered her own programming as well. I try to imagine it as being similar to a human who is unable to vocalize a human language, though they understand it perfectly fine, and can communicate through other means. Except in the cases of the droids it is probably done on purpose by those programming them. And in addition, they may also lack the hardware that droids programmed to vocalize posses, as for some reason I expect that to be different than the hardware they use to play audio back. On the surface, it doesn't make sense, but there's nothing that prevents you from building a droid that way. A decision was made to do it that way at one point, the reasoning may be now forgotten, it may have even been a mistake, or it may have been that they were designed before speaking was a thing droids could do, but it stuck. And perhaps the astromechs are smarter, so people feared a rebellion, should they be able to speak to everybody, thus all suggestions to upgrade them to have proper speaking functions were rejected. If you see L3 and imagine at least some R3 droids having that personality and thinking, you'd see at least one reason for a droid manufacturer not to alter the way they work. I haven't seen the actual scripts, so I may be wrong on that one, but I would imagine R2-D2, Chopper and such, as well as everyone speaking anything other than Galactic Basic, have their lines in English originally in the script, and that's translated later on, in their cases to beeps. Chopper is the only one who says “I've got a bad feeling about this”, and does often, in Rebels.
None of the characters in Star Wars speak English because it all happened a long time ago in a galaxy far, far away... R2-D2 doesn't speak Galactic Basic to increase his dramatic cuteness. I assume the in-universe reason is that astromech droids don't need the expense of having the sophisticated built-in speech circuitry that protocol droids do. They are just expected to get on with the job and not answer back.
I know Peter Mayhew actually did say his English dialogue on set in ANH, which got dubbed over with Wookie growls in post-production. I think it's the three-hour documentary on the OT's DVD release which actually includes footage of him speaking English on set. "That old man is crazy." "You said it, Chewie. Kid, where did you dig up that old fossil?" Likewise, it is BB-8 who says "I have a bad feeling about this" in TLJ, according to Rian Johnson. It's just before Poe makes his first attack run at the start of the movie, BB-8 makes a series of beeps which Leia responds to with "I'm with the droid on this one."
As long as we're talking about speech, why does C3PO talk like a homophobic straight man who's trying to pretend that he's gay?
It is tough to keep in mind but not everyone treats droids as actual people is a bit more unusual in the galaxy. We see Lars being very dismissive, Finn responding with derision at Poe's comment, as well as Han. They are more of an appliance, expected to do as they are told. Given that astromechs are meant to be on ships, do their job, and return to their docking stations.
Well first off the real world reason is: because it's a space fantasy fairy tale broadly inspired by 30's serials and robots that go "beep boo bop" are part and parcel of that. Secondly: Lucas thought of R2 not as a person per see, but more akin to a family dog. Lots of personality, a degree of self aware intelligence, but very limited when to comes to communication. Thirdly: Astromechs aren't really designed to interact with organics much if at all. They're supposed to repair and maintain systems, and serve as what is essentially a removable navigational computer/autopilot on small craft. They can understand organic verbal commands but they don't have any particular need to talk back as the only things they're meant to actively communicate with is other droids and computer systems, hence: droidspeak.