'I really don't care': After THAT Twitter scandal, actor Alec Baldwin discusses fatherhood, working with Woody Allen and being in the public eye

His bust-ups – with photographers, reporters, even his own daughter – have long threatened to overshadow his exceptional work. Now, at 55, can Baldwin still command an audience after his latest, most surprising act of self-sabotage?

There's something about Alec Baldwin that's tremendously imposing. He's broad-shouldered and muscular – but it's not that. Nor is it his ice-blue eyes. Maybe it's the roles he plays, typified by his unforgettable cameo in Glengarry Glen Ross, as the bullish real-estate maestro telling his cowering agents to, "ABC – Always Be Closing". Or, more likely, it's the threat of his infamously volcanic temper, which erupts with entertaining frequency.

Today, he's dressed in black trousers and T-shirt, revealing tanned forearms built like iron girders. In front of him on the table are a bunch of grapes, which he's picking at like a latter-day Roman Emperor. We're in Cannes for the film festival, sitting on a secluded patio of the Carlton Hotel, where Baldwin has arrived to bang the drum for Seduced and Abandoned, a documentary that wryly follows his and veteran director James Toback's attempts to get an erotic thriller into production, and to prove to themselves that the ageing pair still have what it takes to make it in Hollywood. It's a touchy subject.

"I'm 55 years old. I don't give a shit!" he says. "I've worked all I want to work for the last 33 years. Do I do things I want to do? Sometimes…" Not that he has much to complain about, following his seven-year stint on sitcom 30 Rock, which picked up oodles of awards along the way. But acclaim isn't everything. "When you do this for 20, 25 years, you reach that point where you realise this is not the most important thing in your life. If I work – great. If I don't – I'll live."

Baldwin does not enjoy being in the public eye. When we meet, it's just a couple of weeks before his anger-management issues will once again get the better of him. The Daily Mail had accused his wife, 29-year-old yoga instructor Hilaria Thomas, of tweeting about wedding presents, recipes and TV appearances while attending the funeral of Sopranos star James Gandolfini; she strongly denied the claim, saying she'd left her phone in the car.

Baldwin came to her defence, aiming expletive-filled tweets at the reporter in question, culminating with the following unambiguous statements: "[I'd] put my foot up your f**king ass, George Stark, but I'm sure you'd dig it too much," and, "I'm gonna find you George Stark, you toxic little queen, and I'm gonna fuck… you… up."

In reaction to the resulting outcry, Baldwin shut down his Twitter account then sent a letter to the gay-rights group Glaad (Gay & Lesbian Alliance Against Defamation). "My ill-advised attack on George Stark… had absolutely nothing to do with issues of anyone's sexual orientation," he said. "As someone who fights against homophobia, I apologise."

While Glaad supported the apology, pointing out that Baldwin had previously lent his voice to a campaign for marriage equality, it wasn't enough contrition for some: assorted gay-rights campaigners urged chiefs at Capital One bank to drop Baldwin as the face of the company. Admittedly, in the rush to crucify him, it seems to have been forgotten that he donates his fee for these adverts to charity (he's a regular philanthropist, too, supporting arts organisations such as the National Dance Institute and the New York Philharmonic Orchestra).

None of this excuses the hateful language of his tweet. Nor is it the first time Baldwin has been in trouble. There have been tussles with photographers – in 1995 he was placed under citizen's arrest for allegedly slugging a cameraman – and an incident in 2011 on an American Airlines flight, from which he was ejected after refusing to shut down a game on his phone before take-off.

Quite what this latest tantrum will do to his career is hard to say. Baldwin himself seems indifferent about Hollywood these days. "I used to think, 'I want to be in that movie! Why aren't I in that movie?' Then I'd see the movie and I'd say, 'You put him in that movie! You gotta be kidding me! You chose him over me!' It used to drive me crazy. Now I don't care."

He's had a varied career, to be sure – from hosting Saturday Night Live and the Academy Awards to voicing the US version of Thomas the Tank Engine. He seamlessly moved from television soaps The Doctors and Knots Landing in the 1980s to major-league movies such as Beetlejuice, Working Girl and The Hunt for Red October. In 1991, he met his first wife Kim Basinger on the set of The Marrying Man. A year later, he won a Tony nomination for his role as Stanley Kowalski opposite Jessica Lange in a revival of A Streetcar Named Desire. If the mid-1990s saw Baldwin hit a slump, he found his groove again in the past decade. He played Phoebe's lover in Friends, narrated The Royal Tenenbaums, acted for Martin Scorsese in The Aviator and The Departed, and earnt an Oscar nomination, for his brutal casino owner in 2003's The Cooler. Then there was 30 Rock.

In the sitcom, Baldwin played a TV network boss opposite Tina Fey's show producer. It was a big hit, won many awards, and Baldwin is fiercely proud of his work as Jack Donaghy. When its backers NBC cancelled the show after its seventh season, he even offered to cut his salary by 20 per cent. "NBC is in so much trouble that they could not save the show," he sighs. "But it's OK."

Ask if it bothered him and you get a typically frank response. "I've learnt, at my age, that whatever happens is OK," he says. "I really don't care. I wanted to do 30 Rock for [my] lifestyle. The main reason to be in the show was to have a reliable schedule to visit my daughter [Ireland] in Los Angeles. My contract said that I only had to work Tuesday, Wednesday, Thursday, k every week. On Friday, I get on a plane [from New York, where the show was recorded] to Los Angeles and visit my daughter and I come back Monday. This was my schedule for 2006 to 2009 – then my daughter got a little bit older. Now she doesn't want to see me every weekend!"

A statuesque 6ft 2in blonde, who seems to have inherited her mother's looks and her father's temperament, the 17-year-old Ireland stems from Baldwin's marriage to Basinger, which ended in divorce in 2002. She is already a staple of the gossip pages, notably for her burgeoning modelling career.

The father-daughter relationship has not been an easy one. Lest we forget, in the midst of Baldwin's vicious custody battle with Basinger, one that left him at breaking point (chronicled in his book A Promise to Ourselves: A Journey through Fatherhood and Divorce), back in 2007, he left a vitriolic voicemail for Ireland, calling her "a rude, thoughtless little pig". While that went public, she would appear to harbour no hard feelings. On her Tumblr page, she posted: "My Dad has made some mistakes in the past. He has spoken out of place, he has let his temper get the best of him, and he has reacted towards things in ways he shouldn't have… [but] my Dad has moved on. He recognised that he needed to change, so he made changes."

If anything, Baldwin sounds like he's trying to atone for his past with Ireland. The way he describes them, those precious weekends he spent with her sound sweet-natured. "I would drive to Los Angeles and I would drive my daughter to a party, and I would park my car down the street under a streetlight, and I would have The New Yorker magazine. I would have a stack of them, and I would sit and read. My daughter would text me – 'Come pick me up!' I drive back to the party, pick up my daughter. She'd say, 'Take us to dinner.' Her and her 10 friends! Then she'd say, 'Come in and pay the cheque!'"

In any event, Ireland has not put Baldwin off fatherhood; he is about to have his second child, with Thomas. "That, I'm excited about," he smiles.

He's also excited about Blue Jasmine, the new film from Woody Allen, in which he stars. This is Baldwin's second outing with Allen in two years, after starring in To Rome With Love. "I love Woody, I adore him," he says, though this time the draw was the chance to work with Cate Blanchett, who plays Jasmine, "a very troubled woman" suffering from a typically Allen-like midlife crisis. Baldwin plays Hal, her spouse, "perhaps not the best husband in the world".

After spending all those years "in the comedy coal-mine" of 30 Rock, Baldwin admits that he "hadn't been around the dramatic actresses of that level for quite a while. And Woody really put her through the machine, because she's so talented. Take after take after take of very exhaustive, emotional scenes. I sat there at the end of the day and thought, 'She is unbelievable.'"

Cue a discussion about the state of acting today – and how he dislikes what most men are doing. "Most contemporary actors want to seem sullen, virile, lean, and they just bore me beyond any measure. The best acting typically in movies today is done by the women. There are some men who I admire, but not many. I love Colin Firth, I think he's great. I like Clive Owen. I love Leo [DiCaprio], when Leo gets to do a film that I think is worthy of him. But the women are the ones I most have my eye on."

Performing is something he has taken seriously from a young age. Baldwin grew up in a large family on Long Island, and acting was his release from his domestic chores, taking him to the acclaimed Lee Strasberg Theatre & Film Institute. "I was the one who raked the leaves and cut the grass and shovelled the snow and cleaned the dishes," he told one interviewer. "My brothers were like hillbillies. They had no sense of responsibility. They played baseball." His dedication paid off: though his brothers acted, none has been as successful.

Yet these days, it seems he's lost that buzz. "I want my life to be less performing," he says. Partly it comes from the weariness of being in the public eye. But there is also the realisation of how little control he's had over the films he's made. "Sometimes the movies are interesting, sometimes the scenes are interesting, but I've never had a good time making a movie! Not one time in my life. Never!" He cites the thriller The Edge as an example, claiming the director Lee Tamahori "didn't understand the movie". Why not direct yourself? "I would rather stick my hand in a garbage disposal."

At least making his Hollywood insider doc Seduced and Abandoned seems to have enlivened him. "I got to spend time with James [Toback]," he explains. "It was only 11 days, but he's one of the funniest people you've ever met in your life. We didn't make any money, but I don't need money. I've worked and I've made money." So what's next for Alec Baldwin? "My wife is having a baby. That's the project I'm working on now."

'Seduced and Abandoned' is out later this year. 'Blue Jasmine' opens on 20 September

Alec or Jack?

Jack Donaghy, the power-drunk TV network executive Alec Baldwin played for three years on 30 Rock, has the same gift for a quote as Baldwin himself. Can you tell which of them uttered the following lines?

1. "I wanted to be president of the United States. I really did. The older I get, the less preposterous the idea seems."

2. "We are an immigrant nation. The first generation works their fingers to the bone making things, the next generation goes to college and innovates new ideas, the third generation… snowboards and takes improv classes."

3. "I'd never been to an audition before. It was upsetting. A grotesque carnival of human misery."

4. "To be in this business and have tremendous integrity and only make distinguished choices is very tough. Denzel Washington's career is an enormous luxury. Compare him to Wesley Snipes. Do you think that they set out for it to be that way?"

5. "Yes, my daughter is Canadian- American, but I'm going to treat her just like a human baby."

6. "Remember, sex is like a Chinese dinner. It ain't over 'til you both get your cookie."

Answers: 1. Baldwin; 2. Donaghy; 3. Donaghy; 4. Baldwin; 5. Donaghy; 6. Baldwin