Is there any manga in which the female lead protects the male lead. The male lead falls for the girl but is annoyed by the fact that he's a weakling and the girl has to protect the guy.

Or, a manga where the girl takes care of the guy but actually he can take care of himself but acts babyish in front of the girl to get attention. Example- 17-sai kiss to dilemma, ouji sama ni wa doku ga aru.

Try Fushigi no Maria-kun.

Mugen Sowaka - Toukyou Shoujo Onmyouji

I would also recommend the Female Bodyguard category tag.

Nulla in mundo pax sincera
"Always go too far, because that's where you'll find the truth." - Albert Camus

SP x Baby. A light, breezy read that'll be fun if you're not too concerned about shoujo cliches and nonsensical plot development. 😉

mostly angry, always witty

9 Banme no Musashi
Tokyo Crazy Paradise

Chinese Translator Accepting translation requests if the manga interests me!

The main guy is nothing like your requirements, but he does like female lead first (and possessive, which is yum): Grand Sun. I second SP x Baby, definitely fits some of your requirements for the male lead.

I'm in idol hell T.T