pert:
The AT firmware runs at 115200 baud by default. However, SoftwareSerial library does not work reliably at that speed. So you need to use the correct AT command to change the baud to 9600, which is the speed at which your sketch is currently configured to communicate with the ESP8266.
As far as I understand AT commands are used to control ESP, but my problem is -> I can't even "talk" to this module. I mean, it doesn't respond to any commands, not only checking IP/connecting to local network, but whole communication between me/arduino and the module doesn't work. ESP seems to be turned off even if red LED lights all the time on its surface.