An oldie but fits most of your requests
Why why love - Taiwan- Female lead is poor but hardworking girl who gets caught up between two brothers of a big company. Oldest brother is responsible and caring; younger brother torments and teases the female lead (“bad boy who rides motorcycles and disappoints the family” lol)
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Siege in fog - China - Currently airing - male lead is a playboy who forced the female lead to marry him after he falls for her. She had her heart broken by an old lover in school who returns and becomes the male leads right hand man.
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Another oldie, but classic:
Goong - Korea - four way love story, male lead is not quite bad boy, but he’s fairly mean to the female lead at the beginning. She’s not quite a “strong” character though, but still a good old drama.
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Master Devil Do Not KISS Me - China - romantic comedy - male lead is an arrogant first born of a rich family; female lead’s mother passes away and donates her kidney to a rich family with the condition that they take in her daughter and care for her till she graduates/gets married. Second male lead is main guys best friend who is caring and nice to female lead.
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Scarlet Heart (Chinese and there’s a Korean Remake; Korean version I feel plays up the “bad boy” feel more than the Chinese version; Chinese version female is a stronger character) female lead travels back in time to ancient times. Triangle between 4th prince (“bad boy”) and 8th prince.