pert:
You should post a link to this ESP8266_Simple library you're using.
Totally agree, sorry for trouble:
You should provide information on what you've done regarding the firmware on the ESP8266.
Current firmware is "ESP8266 v0.9.5.0 AT".
Also tried firmware used in this project:
https://create.arduino.cc/projecthub/circuito-io-team/simple-arduino-uno-esp-8266-integration-dba10b
It was called "v1.1.1.1 AT". Didn't know if that was official one or just "homemade" (like their library for ESP8266) so decided to rollback to 0.9.5.0 mentioned above.
ieee488:
I found Arduino to ESP8266 By Serial Communication – Martyn Currey to be very helpful.You must buy a 3.3V AC adapter. The ESP8266 is power hungry and the Arduino Uno cannot supply the current that the ESP8266 needs.
Thanks for link, as I see it contains a lot of useful info, will check it when I wake up 
In terms of power for ESP... It's not a big deal to get adapter, but then my question is - if it's required for ESP to work properly, why (at least) a dozen people managed to create working projects with ESP supplied with Uno?