Hello,
First of all, please refer to the very first post of this thread - click on the link in it and you'll find an online spreadsheet with all the answers. Here's the link:
https://docs.google.com/spreadsheet/...3ZnZlb1E#gid=5
Go to the sheet called "AOS Process" to find a step-by-step guide of AOS process and "AOS Package" to see what documents you need to mail along with your I-485.
You will need to include your emails from KCC (DV fee payment receipts and email confirming your intention to do AOS) in the AOS package, as well as your medical examination report, but make sure you include all the other documents listed in the sheet that apply to your case.
The NOA letter is not sent by KCC. You will receive it within a couple of weeks (hopefully) after you mail out your AOS package to Chicago Lockbox.
If your case number become current in May, you may ship your AOS package to Chicago Lockbox anytime now. Don't wait until the end of April. In fact, the sooner you mail it out, the better. Just make sure you have all the necessary documents enclosed so that don't reject it. Your priority right now is to mail it as soon as possible though.
P.S. I don't think you need to worry about a criminal record if you're doing AOS, but you may want to include a copy of your high school diploma, even if you have a bachelor's degree. Make sure all your translations are certified.