fatal error ESP8266WiFi.h: No such file or directory
The ESP8266 core for Arduino that you're trying to use is for programming ESP8266 directly. You are trying to program an Uno with an ESP8266 connected to it for use as a WiFi module. For that application the ESP8266 core is of no use. For your application it's most common to run the AT firmware on the ESP8266 and communicate via serial. If you're doing that I recommend the WiFiEsp library:

https://github.com/bportaluri/WiFiEsp

Since you're uploading to the Uno you should have Arduino/Genuino Uno selected in the Tools > Board menu.