In my estimation a huge part of what American Idol should require from contestants is song writing and arrangement ability, along with singing and voice. If Kelly Clarkson is writing these hit songs with collaboration are others like Cook, Daughtery, and Underwood writting their stuff too? I hope so.
I don't know about Cook and Carrie, but I'm pretty sure Daughtry writes or co-writes some of his songs. Kelly has written or co-written songs on all four of her albums to date. Not every song, but about half of all her material so far. Kelly wrote her Grammy-winning hit 'Because of You' when she was only 16. Not only was it a smash on Pop, but the Reba-Kelly duet version was a smash in the Country genre.
However, Kelly didn't co-write 'Hook Up'. AI judge Kara DioGuardi did, with a couple of other songwriters.
Daughtry writes a lot of his own stuff and Carrie Underwood has recently begun writing/co-writing some songs as well.