Both true, and ditto for Rorschach’s original, The Question.
Originally Thor in Marvel’s Ultimate Universe was considered mentally ill.
It was believed he wasn’t really a god but a delusional hippie environmentalist.
The Sentry was mentioned, though in the end it was revealed that he himself was the Void.
Deadpool is definitely mentally ill.
You can make a case that most heroes (and villains) are insane to do what the do.
Braniac 5 of the Legion of Super Heroes went insane and shot Chameleon Boy and tried to destroy the universe.
I would say Mento went ga-ga at the conclusion of Swamp Thing #50.
John Constantine was in an insane asylum for a while, but I’m not sure you can count him as a super-hero.
There’s Irredeemable an excellent series by Mark Waid that answers the question “What would happen if Superman went completely bonkers?”
Sorry, but no matter how you dress things up, Batman is a deeply disturbed person who chose one of the oddest methods conceivable to deal with his parents deaths. Even DC Comics decided that after almost 80 years the only way that they could portray this character is as a complex man with many dark tendencies.
That’s why the only “light-hearted” portrayals of Batman in recent years have The Brave and The Bold (which had several darker episodes) and the Joel Schumacher Batman’s (which were abortions).
He’s a profoundly disturbed character whose personal quest borders upon an obsession.
I believe it was in the non-canon Dark Knight Returns that Batman called him the most dangerous person alive.
IIRC, Guy Gardner was originally a caring social worker who suffered some sort of head trauma. He became a right-wing, aggressive caricature during the Giffen era JLA.
I must respectfully disagree in part. O’Neil’s question seemed to be one of the sanest, although conflicted, individuals in the DCU. The only crazy I can recall from him was due to involuntary exposure to hallucinogenic drugs and hallucinations when severly injured.
Peace - DESK
Why is he any more disturbed than Superman, or the Flash, or any of the other literally hundreds of costumed crime fighters in the DC universe?
Psychiatrist Doc Sampson diagnosed the Hulk as being a manifestation of Bruce Banner’s multiple personality disorder
Daredevil was amnesiac for a while, and thought he was his father
Superman, Flash, the Green Lanterns etc, don’t obsessively sit in a cave all night every night, and drive away everybody who tries to get close to them (multiple times in some cases), including the people who raised them for most of their lives.
The Watchmen are not Superheroes, though (except Manhattan).
They’re as super as Batman is. And Ozymandias is definitely metahuman, too, even if not to nearly the same degree as Manhattan.
Be fair now, Superman does famously have his Fortress of Solitude/Supercave. Of course, he is an alien, so maybe judging him by human standards of mental health is going down the wrong path. 
…then neither is Batman, Green Arrow, Black Widow, Hawkeye, etc. etc.
Black Canary struggled PTSD in Grell and existential/tormented mental illness in Birds of Prey. Which is to say, she was fucked up, but not insane.
Eh, depending on the incarnation, Batman hangs out or has hung out with the Justice League, has mentored the various Robins, and has various relationships with the rest of the Batman Family (Damn you and your tricks, Batmite!). He and Superman are close friends. He’s dated a bunch, married, had kids of his own. He and Zatanna had a thing. So did he and Catwoman. So did he and Talia As Ghul. So did he and Wonder Woman. He’s got a good relationship with Lucius Fox and a bunch of other colleagues and employees at the Wayne companies. He treats Alfred Pennyworth almost like a father figure. He’s friends with Commissioner Gordon, and has a kind of grudging respect for Renee Montoya, and a shaky truce with Harvey Bullock. He and Two-Face were once friends.
My point here is that, while sometimes Batman is portrayed as an obsessive loner, that’s not always the case. He’s been portrayed as having family, friends, relationships.
Star Boy of the Legion, who was also Starman of the JSA (after Infinite Crisis) was revealed to be schizophrenic in a way that’s treatable by 30th/31st-century medical science, but when he was stuck in the 20th/21st century for an extended period, his mental illness was apparent.
I was gonna post the Badger, but he’s in the OP. Somewhere I have a cover with him punching someone at the supermarket register, screaming “Next time fill out the goddam check BEFORE you get in line!”
So instead I’ll go with Bob Burden’s Flaming Carrot. Having read 5,000 comics at a single sitting to win a bet, this poor man suffered brain damage and appeared directly thereafter as the Flaming Carrot, Champion of Justice, Master of Adventure, and Dreadnought of Chicanery!
The Flaming Carrot Website (use the pulldown menu hidden in the upper left).
Ras Al Ghul means “Head of the Demon” and is a title, not a name. Talia has no surname.