Single Split Air Conditioner & Supply/Exhaust Fan

First Question

Does single split AC (the type with evaporator indoor and compressor outdoor) vent or exhaust air outside and thereby create negative air pressure in the room?

Reason being I am trying to ventilate my home gym room through the door gap with a wall mounted supply fan in the living room outside the gym room, so the gym room needs to have negative air pressure. Installing an exhaust fan in the gym room defeats the purpose since i would like to work out in an air conditioned gym.

Second Question (assuming the answer to the above is yes)

In another room (outside the living room as well), a master bedroom, i am planning to install another wall mounted supply fan but due to space issue, it has to be on the same wall as the air conditioner, right beside it.

Does this then mean that the air vented by the AC compressor unit would immediately be sucked back into the supply fan, introducing the hot and high concentration of CO2 back into the room, defeating the purpose?

Would this then mean that having a bigger CFM supply fan in the living room would be better than having 2 smaller units one in living room and another in bedroom?

As the unit in living room will not be affected by any AC and the air will still be able to travel to the gym room & bedroom via the small door gap provided that the AC is on and is able to create negative air pressure in the rooms?

Thanks.