Best example would be Nagareboshi; the female lead is jaded and she works in sex industry, while the male lead is an upright, responsible guy. Similar-but-not-quite are Keizoku SPEC, Okitegami, Hard Nut, and secondary couple from Wild Romance; basically the girl has some screw loose and the guy is the sane one. If possible I prefer the guy NOT to be too kind, and the dynamic is more like tough love if anything. Thank you!

Note:
doesn't have to be romance, can be just male x female partnership like examples above.
doesn't have to be the main couple, can be secondary one.
I'm open to any kind of recs, be it movie, drama, non-asian movie/series, manga, or anime. 
Have you tried Playful Kiss?  There is also a Thai version called Kiss Me that I liked.  They male lead is straight laced, smart, and a bit cold while the female lead is very sweet and bubbly but does not do very well in school.  Give it a try---I think it fits what you are looking for.
35 sai no Koukousei - the female lead is a trouble-maker and she's kind of a rebel, while the male lead (the teacher) is, like you said, an upright, responsible guy
The Memorandum of Kyoko Okitegami - the girl forgets everything she's done within a day after she sleeps, so her situation is pretty problematic, and the guy is just a sweet, kind person, who always wants to help her
Tokyo DOGS - the girl has some problems and she's protected by two guys who even though they're kind of weird too, they're not bad people
Biblia Koshodou no Jiken Techou - the female lead is very strange and she's basically an introverted, while the male lead is more of an normal, level-headed guy
Otenki Onee-san - a weird girl and a pretty normal guy
Celeb to Binbo Taro - it's about a very rich girl with an arrogant attitude who changes her values after meeting a poor, but kind-hearted man
Take Care of the Young Lady - similar plot as the one before
Ohitorisama - the female lead is an workaholic and has her issues, while the male lead is a shy, but level-headed guy
Zenkai Girl - similar plot with the one above
Q10 - the girl is actually a robot and obviously, she has a werid behaviour and the guy is just a normal, but open-minded person
Orange Days - (OP has seen it, but mentioning it for other readers) - girl totally thinks she can throw tantrums and give up on herself because of a disability. Guy is very patient and kind - hearted, but doesn't baby her. He pushes her to snap out of it.

OP dropped the following dramas, but if you're looking for characters of this particular archetype, consider giving these two another chance:

You Who Came From the Stars - female lead is a diva who kind of expects everyone to kiss her butt. Of course, our alien boy hero is way too cool for that.

It's Okay, Love - female lead is kind of bitchy and constantly tries to sabotage her relationship with the male lead. It's mostly a defense mechanism due to stuff in her past, but he sets her straight.

Other titles:

The World That They Live In (written by the same screenwriter as It's Okay) - again bitchy female lead - even more so than the F-lead in It's Okay. This girl was kind of annoying because she was so entitled and in many situations downright rude. Anyway, male lead knew how to stand up to her foolishness.

Aishiteru to Ikketure - very immature female lead. Male lead is a patient, level-headed guy who tries to deal with her with varying success.

Heart to Heart - female lead suffers from a series of minor mental illnesses, male lead - a 'psychologist' - tries to counsel and 'cure' her.
Boss and Me might also work.  I don't know if the female lead is "problematic" but she is a bit silly and clumsy while the male lead is very cool and level headed.  There is also plenty of "tough love" in their relationship.
Nodame Cantabile - weird af girl and a stick up his ass guy
DoS Deka - sadistic female + her submissive partner. average as far as detective shows go

My Rainy Days - bully and prostitute high school girl falls for older college professor
The Perfect Couple - squeaky voiced slightly rude journalist + grumpy male cop
Sakuran - insolent edo period prostitute who wants to be oiran. masanobu ando's character was the quiet supportive guy
Poison Berry in my Brain - woman has 5 characters in her mind who dictate her actions, the other male lead is the good guy
Why don't you play in hell? - slightly sadistic daughter of yakuza boss + a wimpy guy

anime Black Lagoon - most extreme case, homicidal female(s) and a responsible Japanese salaryman