In the meantime, I've removed the code that "homes" the Z-Axis from my gCode, and I can actually print.  This means (painfully) manually setting up my Z each print, but it's better than nothing!

Thank you both, I'm very impressed by the community already!

Back left of the machine is a black thumbwheel on a screw which is used to adjust starting bed height.

Looking inside the back left of the machine, the bottom of the screw should hit a switch on the rear left of the moving bed just as the bed touches the nozzle.

You can rotate the thumbwheel to adjust the height up and down for fine adjustment, but if this is not aligned correctly with the switch on the bed, or something is broken, the printer may very well be doing as you describe! Watch what this does when you try to home z, and adjust accordingly. Post a pic if you're still not sure.

Solidoodle 4 (fresh out of the box)

When I run the standard gcode for starts, the print bed pushes up to the extractor and just keeps trying to push.  The motor strains for about 20 seconds, while I frantically push Emergency Stop.  I've tried setting the "home" digitally, and by turning the gear on the back of my printer.  Neither seems to fix the problem.  Every time I print, it just keeps pushing the bed up into the extruder, no matter how ridiculously low I "home" my Z to. 

Please tell me I'm doing something idiotic.  I can't find any posts about big Z calibration problems, only little ones measured in tenths of millimeters.