1. Where FL falls in love with ML.  (ideally he should be Cold. Or he had a girlfriend?) And FL has such a sweet character.  She is happy when ML talks to her, touches her (e.g. pats or strokes her head), pays attention to her.  And she falls in love first, of course, and he is so distant from her at first.  She does everything to make him fall in love with her because she really crushes him.  

1.2 It could also be that she is in love with such an idol, they meet by accident and of course slowly fall in love. Her dream came true. 

Itaewon Class is the best drama that can meet your first requirement in my opinion

Backstreet Rookie - The ML isn't cold in this one. But matches the other things you mentioned.
Find Me in Your MemoryThis one fits too.
Nodame Cantabile