In countries like China, Japan, South Korea (the three countries I'm most familiar with), it is quite common, though a few feminists here and there don't like it, that men mention a girl's appearance directly and compliment on it. Girls also like to get complimented, and girls who don't tend to lose her self-esteem.
However, Western countries (most notably Scandinavia if that is too broad, but also countries like Baltic, France, Germany, etc.) are definitely more rigorous on gender equality, and I feel girls don't like to get mentioned her appearance as it might make them feel a sense of being seen as an object or doll.
I also heard that complimenting directly on their appearance is not appropriate but praising what they achieved on their own efforts, such as nails, hair styles, and fashion, is OK. Is this correct?
The compliment is quite direct on their appearance, such as "You are so beautiful!" or "You are so sexy".
I'm mid-20s men and girls I usually talk with are in their 20's. They are mostly those I met on Tinder (on the first date, or during chat within the app), or sometimes I met in public (e.g. a girl approaches me while I'm using my laptop at cafe and starts to talk for a while), and definitely nowhere near to close friends (it is rather at the start of our encounter). They all have a bachelor or master degree (or going to get it).
And sorry but I would like an answer in what you believe they feel, not what you feel. Also, please consider the content of the post, not my personality, if you want to downvote (already got 3); otherwise let me know how I can improve the content.