Does the completion of a lesson stay with a learner even if that lesson is removed from one course and put into a new course?

A lesson completion does "stay" with a learner when it comes to lesson reporting. Once a learner completes a lesson, it will appear as completed in lesson reports. 

For purposes of course reporting - a lesson is tied to a course and course completions are tied to course enrollments, which include new data for the lesson each time it is part of a course enrollment. 

This means if "Lesson X" is included in both "Course A" and "Course B", then completing Lesson X as part of your enrollment in Course A does not count towards your enrollment in Course B. You still have to complete Lesson X in Course B for Course B to be considered complete.