Warning! The only way to change the address USCIS uses for your OPT-related mail is to submit an official change of address to USCIS using their website tools. The U.S. Postal Service can't forward official government mail (like OPT cards) to a new address.
Between 3-5 months from the date USCIS receives your application, though it can be faster depending on the time of year.
You may use the case number on your receipt notice to check the status of your application online. You will need to choose 1) I-765 as the application type and 2) Potomac Service Center (all OPT applications go there). Then look at the chart line that says "qualified F-1 academic student".
No. The 60-day grace period does not apply to students who are waiting for the government to process their OPT application. As long as your application arrived by the government's deadlines, you can stay in the US and wait until it is processed.
You are still an F-1 Student while your OPT application is pending. OPT is a benefit of F-1 status, it is not a new visa type or status.
The answer depends on your situation. If you were issued a new I-20 with the OPT recommendation but have not yet mailed your application to USCIS, please notify your DSO of your change in plans. Your DSO can cancel the OPT recommendation.