I agree with the others that the emails seem to be a bit casual, and I would never in my wildest dreams tell my direct boss that I wasn't available due to plans with friends, let alone anyone higher than him. Come to think of it, if I had lunch plans with friends, and even an equal level colleague asked for my time, I would very lightly make sure that this was the only time they are available, and then cancel lunch if necessary. When I go out to lunch on work days with friends, it's ALWAYS understood that work comes first.
I am really disappointed that I had to continue literature type English courses, when what I really could have used was technical writing and PROFESSIONAL communication classes in college. This is a huge deficiency for me, and unfortunately, I'm learning more as I go along (and leaving bad impressions in my wake).
Also, along with the professional communication subject, you might ask your supervisor if big boss should be copied on lower, group-level emails, and when it's appropriate to do so.