johnwasser:
I'm confused about your reset circuit. It looks like the button will connect the Reset line to +5 but since the Reset is an active-low signal that makes no sense. Typically you would have Reset tied high with a pull-up resistor and use the button to connect it to ground to force a reset.Looks like that resistor might have been acting (through the LED) as a pull-up on your Reset pin? If you let the pin float the processor might be resetting.
If the Z80 is a CMOS chip then all logic inputs must be pulled UP or DOWN. Floating inputs can cause two complimentary transistors to turn on at the same time and thus cause a short circuit between power and ground. That would cause the heating you noticed.
Well, reset is active when the button is not pressed. In order to disable reset, I need to press down and hold the button, I know it's weird, but this is how it works now 
But this clears up your confusion?
By inputs, you mean all 40 pins or just the pins that "take data in"? If the second option, then all inputs are connected. Here's the Z80 pin scheme:
And the CPU itself is either NMOS or CMOS, I'm not sure here.
So, what would be the next step? 