1- you are fine, I wish I was in your position right now, my DAT is next week
Yes, earlier is better, the application opens june 2nd so I'd say june until school starts is a good time to submit the application.
2- yes, you can submit before you take the DAT BUT the school will be like "alright cool, we have your application but we won't look at it until you send us the DAT scores"
3- if you can study for the DAT while taking classes,, great. if you can't then study for it after final exams end in May and then take it july-augustish you should be fine in this timeline.
4- take it from someone who is currenly crying herself to sleep everynight, taking the DAT before orgo II is not the brightest idea I ever had
i hope everything works out for you and that you will do great in orgo