As you can see, I use an Arduino Uno for power supply and as ISP. However, when I connect both the first (VSS) and second (VDD) pin of the LCD screen, the voltage in the breadboard drops to zero (Arduino also stops blinking its leds). I was wondering what is causing this voltage drop? Is this possible because of bad soldering or a short circuit? When I connect VSS without VDD and vice versa the screen turns on. Here is closeup picture of the LCD holes:
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.