Im disagreeing with the other posters and saying yes, and yes.
Just get all your classes in there. Not so much for the minute increase in GPA, but just so they see the additional classes you took and how you did. Gives more evidence that you're a good student (especially if you're a junior and only have 5 semesters on your transcript right now)
Submitting June 20th will make NO difference in when you get secondaries and interviews. You will be right there with everyone else who submitted June 1st. I submitted around that time last year and it took less than 24 hours to go from "submitted" to "being verified" or whatever the terminology is. Then it would have come out and been done shortly thereafter if I hadn't been missing a transcript (long story -AMCAS lied to me). Even when I submitted early July it only took 5 working days to get verified and I still got early interviews, and an acceptance on 10/16.
Schools don't even get the applicants info until late June/early August, they won't send secondaries until July/August and they won't interview until September/October. Submitted June 20th will put you in the first pile and its nothing to worry about.