Adding WiFi to my Arduino Keypad Lock
Can I directly connect an ESP GPIO pin to the base of the transistor where the arduino is connected?


No. You will effectively be connecting two outputs together, and that is very bad.

Instead you will need to add a second transistor in parallel to the first - one controlled by the Arduino and one controlled by the ESP-01.


  Can I put a relay module between the arduino and the battery pack and connect it to the ESP? 


Sure. No problem with that.


  I will connect the relay so that it will disconnect power to the arduino only when powered on. Is this a viable solution for a kill switch?


As long as the kill switch doesn't also shut off power to the ESP-01 - otherwise it will just switch on again straight away...