06 mustang won't start with key now

I am not at all familiar with that bypass module or your vehicle for that matter, but does the bypass have the halo that has to be draped over the ignition cylinder?  Do you have the bypass grounded to the chassis, or did you use the ground out while running wire?  If it is grounded to chassis, the car is reading your key, plus whatever the halo is telling it.  I know you said it is not programmed yet, but it could be interfering with it.

Remote starting the car without the module programmed, should not hurt it at all. Believe me, I've done it a hundreds of times. Not on purpose, of course, but it happens. The fact the padlock is not flashing when you try to start, tells me you might have another issue instead of the factory transponder. Have you disconnected everything to do with the remote start? Have you tested all the fuses, including those under the hood? On the mustang, the starter relay will also be in this fusebox. I cant remember the exact location, but it is one of the grey relays. Did you do any of your work at the SJB.....if so, check all of the plugs. I have seen Fords come from the factory with plugs barely pushed in.

Test the solid green starter wire on the frontside of the SJB. If this tests good during crank, then I would move to the fuse box under the hood. Rather than just switch relays, did you test the pins of the socket. This will give you a good idea of where the circuit is being broke. It will help to determine if it is a problem inside, or outside of the vehicle.