It seems like most people have a different time table for everything, but I think it's safe to say that you definitely want to have your application submitted by June 1st.
When to take the DAT... It seems that everyone has their own agenda. I wanted enough time to study after school got out, but at the same time have my DAT scores arrive at the school about the same time as my app. (DAT scores only take a couple of weeks to get there, while app, takes AT LEAST a month). So I took my DAT in mid-June. I would suggest that you don't wait any later than early July to take the DAT, but that's just my thoughts. I don't think I would take any classes while your preparing for the DAT, unless it's a class that doesn't require much. I was spending about 8-12 hours per day studying for the DAT and there is no way I would've had time for that and a class.
As far as Kaplan goes, I have no idea when would be the best time to take it.
LOR's - I started asking teachers in early February (if I had a good relationship with them). I think this was good because by the time April rolled around, they were getting slammed with letter writing. Besides, it also shows them that your responsible and plan ahead.
Those are my thoughts....