If I draw a rectangle on layer 0, is there any way I can recover the coordinates of its corners later on? I've tried selecting the rectangle and using "properties" and "attributes," but neither give me any information on the rectangle's location or geometry.
You have to explode (Modify - Explode) the rectangle for "Properties" to show coordinates.
Or use a drawing command like "Line - Two Points" with "Snap on Endpoints" on to give you the coordinates of the rectangle's corners in the Status Bar at the left bottom of the screen.
Oh, there is another possibility: Plugins - List entities. Works on unexploded rectangles, too. Even on multiple entities.
If it is the dimensions, not the coordinates you want to know, use Info - Distance Point to Point and click on diagonal corners with Snap on Endpoints on. The values are shown in the command line.