Well, you applied for OPT on June to request a start date from July. That's the problem.|
From USCIS or your school international student office website, they will tell you that the regular processing time for USCIS to handle a case is approximate 3 months. That is, you will not get your OPT approved before Sep or Oct.
OPT start date is decided by the later one of your request time and approve date by USCIS. Your request time is July and OPT approve date is Oct. So your OPT will begin from Oct.
However, OPT expire date is strictly limited by your I-20. Actually, the valid start date for you to request for OPT is within 60 days after you graduate. So the latest date you can request will be on July. According to this, I-20 your school approves will start from July and end after one year. That is, your OPT expire date is July next year. And you have no way to request more time for current OPT but to apply for OPT STEM extension if you are in STEM major.
The start date and expire date are both on your EAD card and between those date is your valid working period. 3 months you waste is due to your late application by your fault.