At a time when Hollywood is tackling the predatory sexuality on which the entire movie industry appears to be built, some of its favourite actors have made a film about the same issues, but set in a far less prestigious town. Three Billboards Outside Ebbing, Missouri won the top prize at the Toronto international film festival for its depiction of a mother, played by Frances McDormand, fighting to bring her teenage daughter’s rapist and killer to justice in a rural backwater.

Directed and written by Martin McDonagh, of In Bruges fame, the film is as bleakly funny as it is harrowing. Sam Rockwell plays the dumb, bigoted cop who initially appears so infuriating that you almost wonder if he is the guilty party, only for his character to travel in quite a different direction by the film’s end. Such is the subtle power of his shape-shifting transformation that there are now whispers this might be the film that finally sees the undersung Rockwell get noticed by the Academy, and be nominated for an Oscar. Is he excited about that?

“It’s very thrilling,” says the 48-year-old American actor, who has almost three decades of work to his name. We are sitting in a London hotel room, where he has gallantly ensured I have the most comfortable seat. “I’m hearing a lot of stuff – we’ll see what happens. We really worked our hump off on this film, so it’s very exciting – but really it’s an excuse to to celebrate that with free booze.”

His red shirt is brightly patterned; his hair has grown back to its usual scruffier length, compared with the “almost military” cut he wore as Officer Dixon. With his fun-dad-on-holiday aesthetic, he couldn’t look less like the “rednecks, racists, ex-Confederates. Cowboys. A KKK guy!” he has spent years playing. “I’m playing so many hicks and it’s not where I come from. I’m really a concrete creature,” he says, “but they keep throwing me on a horse.”

Rockwell grew up in inner-city San Francisco, brought up by his father after his mum left home. He went to a performing arts high school where he met his best friend, Leroy, after the pair realised that they were both giggling in a lesson “because we were both high”. Leroy, an African-American, took the young Sam to parties where he had to learn how to dance to impress girls, a scene he vastly preferred to hanging out with the white kids at his school, “the Nazi youth,” as he calls them. This explains how he became “a damn good hoofer” – chuck his name into YouTube and you’ll find endless fan medleys of all the film scenes in which he dances.

Indeed, a friend who lives in Brooklyn tells me she once spotted him at a public Soul Train dance-off competition. “She did?” he says, before recalling it. “Oh, yes, I was with Marisa Tomei! We were drunk, it was a blast. I think they gave us celebrity nepotism so we came second. It was a funky after-hours place – you know, south London 20 years ago kind of thing.”

He says he likes going out drinking, but can’t find many dancing spots like that nowadays. “It’s all techno in the clubs now – I can’t stand that shit. Ugh, it’s awful,” he says, looking quite disappointed with the modern world. He shares an East Village, Manhattan, loft with his long-term girlfriend, the actor Leslie Bibb, and rides a lightweight 10-speed bike that he has to lug up several flights of stairs. Unless he takes the subway, where people do sometimes spot him, “but I do pretty good, I shift between the cracks a little bit. It’s funny when people recognise you though, because they look very pensive and sometimes a little angry” – he furrows his brow comically – “like they know you, but they don’t know where from, so they’re trying to figure it out.”

This is surely because Rockwell is that guy who has been in everything, yet somehow isn’t a household name. His big breakthrough was arguably Confessions of a Dangerous Mind, the Harvey Weinstein-produced, George Clooney-directed film in which he played the TV gameshow host who became an unlikely CIA assassin. Then came Matchstick Men, Choke and Seven Psychopaths. He played the racist convict in The Green Mile, the flamboyant Zaphod Beeblebrox in the film of The Hitchhiker’s Guide to the Galaxy and the solo, struggling astronaut in Moon. We don’t see his love life in magazines, even though he was rumoured to have dated Drew Barrymore, his co-star in Charlie’s Angels.

He and Barrymore share a dramatic scene filmed in The Chemosphere, a futuristic hillside, spaceship-style house in Los Angeles, and I ask if it brings back memories every time he drives past it. “Well, I have a lot of memories all over LA, a lot from being a broke struggling actor, too, sleeping on couches. Auditioning for ER, Noah Wyle’s part, I think, and turning up in a white lab coat that I’d found somewhere, to make me look more like a doctor, and holding a hypodermic syringe.”

Sorry, I ask, you turned up to the audition in full costume, clutching a syringe? “Only the plastic part, it didn’t have a needle,” he stresses. “They did think I was crazy though,” he adds, looking mortified.

I ask if he is limited by being a character actor; a category that can impose a cap on success.

Agreeing to an extent, he points out that he has also been a leading man several times, “although I’m never gonna play a traditional romantic lead”. He says character actors do get noticed – just very slowly. “Gary Oldman finally got nominated for Tinker Tailer. Jesus Christ, that guy’s been doing it since State of Grace, Sid and Nancy – he was the guy we all looked to, you know? The fact it took that long for Gary Oldman to get nominated for an Oscar!”

I ask how it feels to be promoting a film about exposing rapists, at a time when the film industry is exposing the ones in its own midst?

“Yeah, the film has a lot of timely themes, doesn’t it? It feels to me like there’s a lot going on in this film, even though it was written before any of that stuff. Trump, Weinstein, before the Ferguson thing as far as [police] racism goes, too – [Martin] wrote it eight years ago. But I think, because of what’s been going on in America lately, it’s just shedding a new light on that, in a way. This feels like The Year of the Woman to me – Wonder Woman’s a hit movie about a woman with sexuality, but she’s strong and she kicks ass. Directed by a woman. And then I think Greta Gerwig just directed a movie that’s supposed to be great, and there are a few others.”

But it must feel like Hollywood is in crisis as well?

“How do you mean?”

Well, the Weinstein Company.

“Well, they’re in crisis, for sure,” he says. “I don’t know about Hollywood.”

Obviously, Rockwell himself is an innocent bystander, accused of nothing. But when I try to get him to acknowledge that men in the Hollywood system have perhaps benefited from a culture that keeps women down, he isn’t overly convinced.

“The allegations are awful, and it’s brave of these women to come forward and talk about it,” he says, sounding a bit formulaic. “It’s a horrible thing. I only had one encounter with him [Weinstein] years ago and I didn’t see any of that.” But how does it feel to have had your big break from someone who destroyed so many other people’s careers? “Well, Confessions [of a Dangerous Mind] didn’t do very well. Harvey dropped it like a bag of wet hammers after it didn’t get a Golden Globe nomination. It was George Clooney who fought for me to be on it,” he says, somewhat defensively. “It feels good that I got to play the part.”

He is shooting Adam McKay’s biopic of Dick Cheney, in which he plays George W Bush. “I’m watching Dubya now,” he says, “I’m watching and listening to him constantly and I find him very likable, I have to tell you.”

Have you looked at his art? That’s my only guilty Dubya pleasure.

“Of course! It’s not bad! He’s very charming. I mean, in spite of Iraq and everything, Guantánamo …”

They’re quite big things, I say – I imagine they horrified you at the time.

“It’s pretty horrifying. The impression I get is that he really thought there were WMDs, and he felt bamboozled when there weren’t, but he’d already gone in there. There’s a real sense of regret in some of the interviews I’ve seen – that he really had thought he was doing the right thing. You do have to develop empathy to play someone, because otherwise you judge your character.”

As for being told he had the face to play him, he laughs. “It’s funny. You’re the canvas, you have to take it. I never thought of it myself, but when the phone call came, I was like, yeah, that does makes sense actually. Especially opposite Christian [Bale] as Cheney. And Steve Carell being Rumsfeld – that’s great!”

It must be interesting to work in a gang when one of his most lauded roles was as Sam Bell in Moon, an astronaut who is entirely alone apart from a robot. The film was directed by Duncan Jones, and Rockwell’s performance was described by the New York Times as “intriguingly strange”.

“Yes – that was a very isolating experience,” he remembers. “We shot it during the writers’ strike so Shepperton [Studios] was a ghost town and consequently we got this amazing crew who were not working – people who worked on the original Alien, a set designer who had worked with Ridley Scott. We got lucky with that movie, it just came together. I think of it as an extended episode of the Twilight Zone. Or that British show, you know, where the prime minister has to fuck the pig?”

Oh, that wasn’t a show, that actually happened.

“What do you mean, that actually happened?!”

I tell him that David Cameron was forced to deny putting “a private part of his anatomy” into a dead pig’s mouth while studying at Oxford, before realising that Rockwell is in fact referring to a curiously prescient episode of Black Mirror.

“Get out of here! That didn’t happen! Come on!”

Moving on – does Rockwell ever worry that he is going to run out of the meat inside him to act with?

“Never!” He barks. “I guess that’s why Daniel Day-Lewis goes and makes shoes or something,” he says, slightly amused. “I could never do that. Or why Sean Penn goes to Haiti. I guess you have to refill a little bit – but I’m kind of a workhorse. It takes a toll. It took a toll on my friend Phil [Philip Seymour] Hoffman. He never wanted to phone it in, on stage especially. He was a dear friend and I think he demanded an awful lot from himself.”

Was Hoffman’s death a wake-up call to step back?

“Well, it was a lesson – you have to have a giggle at the end of the day. This film is important to me, it’s important to Frances [McDormand], because it needs to be life and death when you’re doing it, otherwise nobody’s going to give a shit unless it’s important to you. But you can’t take yourself too seriously. You’re not curing cancer.”

• Three Billboards Outside Ebbing, Missouri will be released on 10 November in the US, 1 January 2018 in Australia, and 12 January 2018 in the UK