
It's omitted in the manhwa all together, but in Eggs and Benedict (Kyle and Nick's novel aka prequel to DTR), there's a scene where Karlyle mistakens Nick and Ash to be together when Nick was supposed to be with Kyle. So, Karlyle scolds the two, resulting in not that great of a "first impression" on Ash.
As for the help from the doctor in the beginning, I'm not sure myself since Eggs and Benedict has not been translated into English yet. It is pretty much like you said though; Ash agrees to help Karlyle out based on a favor he owed from Nick. The doctor probably asked Kyle to help, which then resulted in Nick asking Ash about the favor, and then Ash agreeing to help. This is all speculation though since I didn't read the novel, so take it with a grain of salt.

Kyle and Nick's story is currently untranslated. Here is the link from Google Books: https://www.google.com/books/edition/BL_%EC%97%90%EA%B7%B8_%EB%B2%A0%EB%84%A4%EB%94%95%ED%8A%B8_%EC%99%B8%EC%A0%84/eeCNDwAAQBAJ?hl=en&gbpv=0
DTR novel (translated) can be found on @the.road.to.eternity on Instagram.
am I the only one who feels like we are missing a chapter in the very beginning. when when the doctor said he has someone who can help and between Lyie and ash meeting. like we never know why or how Ash was asked especially when we find out later that him and nick weren't even talking much to begin with. and Ash kept saying that him and lyie first meeting wasn't the best but I was like besides the new years kiss this is the first time you guys are meeting....
on another thought I wish we had a story for Kyle and Nick!!!!!!!!