I'm assuming your question about the fake documents is about how Kid got Conan to Singapore without trouble at the airport, right? I haven't seen the movie yet, but maybe Kid just put Conan in the suitcase and gave him to the airport workers to put on the plane (I saw awesome fanart of this haha: https://mobile.twitter.com/BakaThief/status/1098994898047512576/photo/1)
And probably Kid gave Conan the spray on tan because a mask may not last as long...IDK. I don't think it's a permanent thing (as the one shot in one of the trailers shows Conan back with his normal skin color), but as I write this now, I realize that whether Kaito gave Conan a mask or tan, neither of them might last that long LOL.