and you will a LOW state before you can put the output HIGH when the Arduino starts,
No when it starts up all pins are set to be inputs, therefore they are high impedance and not pulling low.
how can you calculate the 220R in the input
That is just to provide a bit of a resistance that will drop any excess voltage caused by any miss match in the supply voltages of the tow units. The value is chosen so that it is small enough to allow the input pin to still pull down to a zero but big enough to limit the current flow for a miss match. I thought 200R what would you choose? There is no right answer only wrong ones.
You have to know the input current of the play pin to guess the needed resistance to reduce 0.5volt.
No you can guess without any knowledge
Seriously though the resistor is not to drop the voltage it is only to limit the current flowing in any protection diodes of the input device.
If you want some isolation the an opto isolator between the Arduino and the device would be the way to go rather than the half way house of a transistor. With an opto isolator you DON'T connect the grounds together, anything else and you do.