As far as what ESP8266 board to recommend, it really depends on how you want to implement the ESP with that base unit. I would want to put the ESP8266 into that device's enclosure and steal power from it to power the ESP. It depends on how much room there is (if any) and if there is sufficient power available.
If you need to use an external boxed ESP8266, then you need to think of how you want to wire the signals to the external box with the ESP. You would need to power the ESP, so the question is what would you prefer to use to power it? Lots of people use USB chargers. I would prefer an AC-DC adapter with a wired cord, no usb connector. It really is a matter of choice.
Because you need to monitor 4 inputs you will likely need a module based on the ESP-12 module. A board with the ESP-12 and a USB to serial converter would be the most convenient for development. Popular are NODEMCU patterned boards, or Wemos. Lots of others to choose from.