Yes, your DDE card starts from the date you order it. I looked at mine a few days ago (getting ready for next trip), and the expiration date printed on it is the last day of the month in which I ordered it. It looks like they gave me that little "grace period" till the end of the month. If you want your AP and DDE to run very close together, you can get the AP voucher and then order the DDE a few days before your first trip--pick up the temporary DDE letter at EPCOT Guest Relations when you arrive.
It is not like AP where you can buy the voucher but it doesn't start till you activate it.
Yes, the requirement to purchase the DDE is a valid AP at the time of purchase---so if your AP is going to expire, you can still use it to purchase a new DDE right up to the day it expires!