I saw this website that managed to connect the ESP8266 ESP-01 and Nokia 5110 LCD together. And also uses Arduino IDE for ESP8266 and ArduinoJson Library.
This is the best example i've found so far to work on the modules with GPIO broken out.
It is simple bit banging of SPI, which gets around some of the issues using the HSPI port, and pin 15 being tied to ground, pullups etc.
https://github.com/esp8266/Arduino Arduino IDE for ESP8266 (actually compiles ESP code/firmware, using the ESP toolchain)
Arduino Playground - PCD8544 There is no library as such, just paste one of the examples
https://github.com/bblanchon/ArduinoJson Json parser library that compiles in the ESP version of the arduino IDE
With those three, and the "WiFi Client" example, you have the basic building blocks to pull data (eg weather, bitcoin price, stocks) from the internet and display on a small device.