I don't think there is anything inappropriate about asking LOR from this professor. Since you mentioned that his reputation is rising, it can do a great deal of good for you if you get him to write LOR.
The key is to plan your classes well in advance and make sure that you teach all the important chapters that are essential for students' educational progress. The course material always has a lot of topics to cover, but as a teacher it is up to you to decide what to teach and what to skip. At the end of the day, students shouldn't feel burdened.
Professors have a lot on their plate. They tend to misplace things or as in your case they might have misplaced your paper somewhere and are unable to find it. With so much to do in limited time, professors don't like to waste their time going after that one misplaced paper. So if they ask to mail an electronic copy, it is completely understandable.
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It is true that rotation students don't get the expected results cos of very limited time. Most PIs understand this and are little easy on the rotation students. No matter you achieve results or not, it would not affect you when you join a lab. Most rotational students sail on the same boat.