Ok, the lead screw and nut I ordered has arrived. The good news is it looks like it will work just fine for what I am doing although the shaft connector that came with it to go from the 8 mm shaft to the motor is too small for the motor, which has a 1/4 inch shaft. But, the shaft coupler I already had for the old lead screw is just the right size to match up with the other coupler so, with a cut off 3/16 inch screw, I was able to get it all together.
The screw turns out to have 3 starts and a 2mm pitch which results in 6mm travel per revolution. I will probably drive it in 1/4 steps from the EasyDriver so it will result in 800 steps per rev and so 0.0075mm per step. The step interval should be around 300 milliseconds, depending on what the actual final dimensions of the door are.
I should now be able to begin fabricating the door itself. I have a piece of 3/4 inch oak for that purpose. The working drawing is below. I will hinge the door on one side and will also hinge both the nut on the top and the motor on the bottom. I believe I have the formulas worked out correctly to compute the step interval to get it to follow the sidereal angle of the earth. I will do some testing after I have the structure built to see how accurate I am. The final proof will of course be in the pictures I take.