Would I be able to keep the cover on using the installation bracket ADT-VA133 for the AVIC-D3?

dont quote me on this but I believe that you will have to redrill the holes and push them back to have it fit and still close the cover. Now I dont have that bracket but I did have to do this with the one that came from crutchfield.com. I would assume that they would be pretty close in design.
you will not have to loose the cover, just closing it would be the issue

http://www.scionlife.com/forums/viewtopic.php?t=165675&highlight=avic

Yes, you can keep the door...use this guy's install...I installed mine this past weekend and it works perfectly...you can find more info on the e-brake bypass on avic411.com...that worked too...