Complimentary parking is for guests entering a park for a short period for the purpose of dining only. Guests pay the parking fee at a toll booth. After dining they need to go to Guest Relations and present the meal and parking receipts. The parking will be refunded.
When going to a resort to eat, just tell the security at the entrance booth you are eating at (restaurant name). They MAY ask you for a dining reservation number. If you don't have it, they may not allow you to enter.
We have had them require the PS number while special events are on, or during holidays.
Security may give you a ticket to place on your car dashboard. They generally allow three hours time of parking.
I am sure DDE members get free valet parking. Just make sure to tip.

And yes, handicap parking does allow free valet parking too at the Disney resorts. (I don't know about S/D.)
We have not used valet parking, so can't help you on that.
And the S/D does not participate in DDE.