So first off, I'd like to just say that I'm not Japanese nor do I understand Japanese. Perhaps that's why I didn't really understand this part. But anyways when Koji goes over to YUI's house for dinner the first time, he pulls out this piece of paper or some sort of add or something. What is the paper? He said he wanted pay for it(the paper) for YUI. And also how did she end up getting her song recorded, cuz it just kinda showed YUI and her family going to recorded it?

I don't remember exactly since I haven't watched the movie in a while. But I believe the paper is an ad showing that you can pay to make your own record.

Kouji would like Kaoru (not YUI) to record her song, so he did lots of part-time work to earn enough to pay for the recording session. In the end, the whole family, plus Kouji and Misaki (I think) went to the recording studio with her to record her song.

^ I second to what solidsnake said. Basically that's it~ Hehe

Thanks for the clarification...I kinda thought that was it, but just wanted to make sure. Thank you!