This is the problem testing the 2 wire ground bypass. You don't know if that worked till you drive around, and sometimes it can take many miles to lock you back out.
You had a stock non-updated unit, it would have needed headlight flashing before the update, now after the update the 2 wires grounded.
An odd thing I encountered when trying to follow the instructions was that couldn't find the "cell phone mute" wire on connector two, which was referenced in the SMinnTech.com article.
Was it this article?
http://www.sminntech.com/z1flash.html#newbypass
Cause I said: Personally I took the "cell phone mute" wire out of "connector 1" 
Guide on is in connector 2, cell phone mute is in connector 1. But I don't come out and specifically talk about where the guide wire was, just that either one you pull will end up in connector 2.
The odd thing was that the stereo shop had connected the light green wire to a darker green wire behind the dash with a butt connector. If you look at the wire underneath the center console near the actual e-brake handle, however, it's still light green.
The vehicle is a 2005 Ford Mustang....
That is a little weird, but I'm sure people have done weirder things. But the light green wire in the Z1 is the parking brake sense wire, and by grounding it you should be good to go. Other than make sure the other end doesn't short to anything, I wouldn't worry about the color of the wires left in your car.
So now when I start up the car, I still get the parking brake lead error, but everything seems to work. I can watch a DVD, plot navigation, set the time and change the background without issue. This was only during a 5-minute trip around the neighborhood, though.
Should I be seeing the parking lead error at all? ...or is it normal to see it when you first boot the unit up?
No, this is only because you tried bypassing the Z1 incorrectly at one point... (using 2 wires grounded when you should have been flashing your headlights). If you did the correct bypass for the correct way the unit was at the time, that would have never come up. Now that will come up every time till you:
Just hitting the reset button will usually take care of the parking brake error message if it hasn't gone away.
The recessed button to the right of eject. That will make it go away, and I don't think it ever will on it's own and can effect the bypass. So you want to do that.
With this done now, is there any chance the messages can "come back" as others have complained about, assuming the wires stay grounded? I crimped on eye lugs and secured them to the metal frame behind the radio area with lock washers. I doubt they'll ever come loose.
Come back? Generally, if you did everything correct, no. And you talk about the lugs, metal behind the frame, that kind of thing. It's far more likely your problem, if you will have one, will be in the insertion of the terminal end into connector 2. Most people can ground a wire, but the real problem is people either damage the terminal end when they extract the wire or they don't know the correct orientation for insertion in to the hole in connector 2 and that wiggles loose. If that pop up message comes back focus on the wire in connector 2, I'm sure you did a good job with the other end of those wires.
Thanks for all the great info. This is truly a sweet forum....
Thanks. You are welcome.