Good day everyone,
in case someone faces the same issues I did:
- controlling multiple relays from one arduino and sharing the same power (powering relay board from Arduino
- controlling relay which switches high loads circuit causing connected LCD to show garbage and again powering the relay from Arduino...
...the solution proposed above is the only one which helps - provide separated power supply for the relay boards. Once this is done with an extra little bucket power source for 5V (3W) all the issues with relays not working, domoticz not switching relays, LCD going crazy - all solved. I should have done this long before.
P.S. For those, who still do not follow, the relay boards for Arduino contain extra jumper connecting VCC and JD-VCC (plus sometimes extra GND pin). So remove the jumper and connect the standalone, not common sharing 5V power source to this extra JD-VCC and GND (or GND near the IN1 pin) and disconnect the GND from Arduino and then you have a full optosiolation and then there are no interferences and other garbage breaking the Arduino functionality by switching relays coils. (also explained here: https://forum.arduino.cc/index.php?topic=470134.0)