Hey there,
If it were me I would stick to the April date, but once again we all are different and have different study habits so if you think the June date will give you few more days, go for it. One very common mistake that we pre-meds make, and I have made it too, is to think that few more days will help me boost my score a lot. In the bigger scheme of things, and for most people, 10-20 more days won't increase your score drastically. Now, since you have already scheduled a June date, make those days count. Do practice questions like crazy and do a superficial review. Just have a game plan for what you gonna do in those extra days; I've done the same thing in past but didn't do anything significant in those extra days and then regret it late.
As far as application goes, I would send everything in before you get your score. The sooner the better. DO NOT WAIT FOR YOUR MCAT SCORE. Also, try spending maybe 2-3 hours a week starting this week getting everything together/doing research/familiarizing yourself with the process. Do your research, ask people, make a checklist of everything you need to do. Just do a little bit every week (such as personal statement, answers to common questions typed up and proofread) so at the end when it is crunch time you won't be overwhelmed.
Throwaway schools would be a school that is your plan C or D. For instance, a school so far away or is not very recognized that you may go there as your last resort. ex: A California student applying to Uni of South Carolina, Greenville, just for the sake of applying. I personally don't believe in that, I respect every school I apply to but there is no shame in having a plan B and C. I am not the best person to tell you how long it takes, but I've seen 2 days (very early in the cycle) to 3-4 weeks (peak time). AAMC is currently working on making the process faster so hopefully, in 2018 we wouldn't have to wait that long. I hope this helped, lemme know if you have any other questions. Good luck!
