To answer part 2, discharge summaries are required in certain instances, so you may want to check the bylaws and/or policies that govern the hospital for when the physician is required to do so, as well as federal and state regulations. Our bylaws require a DC summary on any inpatient.
As far as the admit note goes, you have to have an H&P to admit a patient, or you can use an H&P that was performed within the last 30 days as long as it is for similar circumstances, if an updated written/dictated admit note is done to document changes in hx and exam, unless this practice is not permitted by your bylaws/policies that govern your facility. BUT, it is allowed under federal regulation (I am also the Medical Staff Coordinator for my facility so I do have first-hand knowledge of this).