Kari,

1. (((. I don't know what goes wrong, maybe timings or my circuit, but instead of having just "Hello world" once, it writes many times and the display is full of grab.)))

To isolate the Reset problem from the LCD software try to modify the routine witch puts the “hello word” to the display. I mean you have to instruct the processor to call this routine after reset when you push down a pin ( INT0/1 for ex.)
If problem with the LCD persist then the reset is god and LCD routine is wrong.

2. ((((Here we come to this reset matter, when reset button is pressed, it works differently than if I simply cut the power for few seconds )))

I suspect the reset capacitor does not discharge completely at power down. I advice you to put a 1N4148 diode ( the most common) between the reset pin and the Vcc pin with the cathode to Vcc.
So when the Vcc goes to 0 V the diode will be directly polarized and will discharge the capacitor.

It is also an other problem: if the power supply is to slow at Switch On ( it has a big output capacitor --- I mean is a god one!!!) you must also increase the reset capacitor so the reset pin comes to the Vcc level several milliseconds AFTER the Vcc pin is stable.
Of course you may consider the RC reset constant time so you can obtain the same effect by increasing the resistor.

Best regards,

Cristian Ceausescu