SM Entertainment is the label that is mostly known for pulling foreign talent for a lot of their songs. The Stereotypes who are known for Bruno Mar's biggest singles like "Finesse" and Cardi B's latest track "Please Me" and have produced some great tracks for groups like Red Velvet's "Bad Boy" (https://youtu.be/J_CFBjAyPWE) and "Kingdom Come" (https://youtu.be/vJEAMow2A4o).

A lot of these songs are originally written with demos in English.

f(x)'s "No More" was originally Ariana Grande's track "Boyfriend" Material" but didn't make her first album.

f(x) - No More: https://youtu.be/6cX0OMtIFkc
Ariana Grande - Boyfriend Material: https://youtu.be/EE79a6FX4ec

f(x) - 4 Walls: https://youtu.be/4j7Umwfx60Q
Original English demo: https://youtu.be/vFuU4rIZZ30

Red Velvet's "Red Flavor" was originally written for Little Mix.

And when it comes to home-grown composers and writers, I don't think you're wrong at all when you say that they are very good at imitating styles. Aside from the hip-hop & R&B sounds we've talked about, they also do a really good job of recreating other sounds from countries they try to promote in.

Some examples of K-pop groups and their Japanese songs that utilize J-pop styles:

Red Velvet - Aitai-tai (https://youtu.be/VqcrnLZkK60) (Nakata-styled electro-pop akin to Kyary and Perfume's sound).
GFriend - Memoria (https://youtu.be/YTWwz6R6jy0) (anime-styled melodies and progressions)

And then you have some K-pop songs that borrow from J-pop like this K-pop group's Korean song:
Dreamcatcher - Fly High (https://youtu.be/39yeTdIuKJU) (very anime J-rock influenced)

And I just learned that one of my favorite K-pop songs from last year was partly written by a Japanese writer (formis9's "Love Bomb": https://youtu.be/-SK6cvkK4c0).

One of the biggest K-pop idol contest/survival shows last year had their main premise revolve around having south Korean K-pop trainees competing against/with members from AKB48, Japan's biggest idol pop act, so it just goes to show how big K-pop has gotten where AKB48 is willing to have their idols participate.

Another thing I forgot to mention last time was that a lot of these newer K-pop groups have members from other countries too. The biggest girl group in Asia (and probably the world) has 5 members from Korea, 3 from Japan, and 1 from Taiwan. Other groups have members from countries like Thailand or ethnic Koreans that have grown up abroad (Canada, New Zealand, Australia), and so they are very much set up for communicating and appealing to foreign audiences.

We also had something kind of interesting last year where a Japanese porn star (Yua Mikami) made a K-pop group (but all other members are also Japanese porn actresses), and they actually did a great job of creating a very K-pop sound with their single: https://youtu.be/LAOT6JscCVU