Despite a wealth of tutorials for setting up and writing code for the ESP8266 WiFi module, there has not been much of anything on programming this cheap wireless module with the Arduino IDE. Finall… ...

There is no reason to use an ESP8266 AND an arduino. The cpu on the ESP8266 is a 32 bit risc running at 80 Mhz, the arduino is an 8 bit AVR running at something less, maybe 32 Mhz.