Just to get this clear:
You want one ESP8266 which is physically wired to a door sensor and a DHT22 temp/humidity sensor.
You want another ESP8266 which is physically wired to a servo and display.
If that is the case, then I would still suggest the "web" solution in post #13 and further, I would suggest integrating the ESP8266s in your existing wireless lan if the location of the 'door' ESP8266 and the 'servo' ESP8266 are within range of this.
Your door sensor/DHT ESP8266 would periodically send HTTP GET command of the format http://192.168.1.20/myRoot?door=1&temp=19&humidity=55. Your server would receive these, parse out the data, and act on it accordingly i.e. displaying the data and operating the servo.
Of course there are other ways of doing this. It is a feature of this forum that some threads catalyse intense discussions about valid alternative solutions and their various merits, sometimes in some considerable technical depth. Often, with the risk that the subject of the original post gets almost forgotten in all this.
The RF solution, which would also be valid, requires two radio devices. If you've already got ESP8266, that is an additional reason for staying with the ESP in this case.