1602a LCD - Voltage drops to zero when both VSS and VDD are connected
You have errors in the wiring. Don't connect display VDD to ground, but to 5V. That error has caused excess current to flow out of AVR IO pins, via LCD input protection diodes to ground. Also, don't connect the contrast pin Vo directly to ground either - just via resistor to ground, or the display will be too dark, just black squares. And finally, there is no current limiting resistor on the backlight, most likely there is no internal resistor and it needs external one or it will draw too much current and might break something.