Is it okay to short IN and GND pins of relay module with controller ground?
It is not okay. The base-emitter of an NPN transistor acts as a diode. That means pin D1 is shorted to GND via a "diode". You need a resistor to the base of the transistor.

The relay module has already a transistor to drive the relay and also a flyback diode. The control signal that is needed depends on the relay module. The explanation "5VDC to 12VDC (some boards may work with 3.3)" on the page of the relay does not sound reliable.
The relay module on that page is not the module that you have. You have a different module that is activated with a low signal.
Nevertheless, I think that a transistor can be used when the IN and GND of the relay module are connected to the collector of the transistor.

Could you please not try to make the schematic as small as possible. A schematic should explain how everything is connected.

Could try to connect the collector only the IN pin? As in the schematic below.



In your circuit diagram, the mains "neutral" is connected to an earth ground symbol. I hope you are not going to connect a mains wire to ground.



When the IN and GND are connected, then the resistor to the transistor needs to be lower. The ESP8266 can output 12 mA. Let's use 10 mA for safety: (3.3V - 0.5V)/10mA = 280 Ω. I suggest using a resistor of 330 Ω or more.