"Has Joker ever been shown being with a woman?"
Uhhhhh.........Harley Quinn in the corner looking pissed off. Then again, she's also depicted as being Poison Ivy's love interest. So, maybe you didn't think of her because of that. What's even weirder is, Poison Ivy has been shown to flirt with/implied sleeping with Batman, Catwoman has been shown to be sleeping with Batman. Hell, even Joker has said some things that make you think there's sexual tension between them. Yet, I never remember Harley Quinn ever trying to sleep with Batman. I know she tries to sleep with Nightwing, and gets horny when she see's his butt, but I don't remember her ever looking at Batman that way, and she's the only female I can think of that doesn't want to sleep with Batman. Even Barbra Gordon/Batgirl got it on with Bats.
As far as Jokers backstory.....yes, technically. There were 2 movies about Joker. They suck. Nobody considers them canon. Not because we don't want to know Jokers history. It's just we refuse to put any weight behind THAT being his backstory. He's a depressed failed comedian with a medical condition that makes him laugh. There, I just saved you about 4 hours.
I like to believe that Joker has no backstory, and any backstory given can't be trusted as reliable, because the narrator (in this case Joker) is unreliable as a narrator. He's likely to just make anything up, and the words cannot be trusted. Would you like to see how I got these scars?