You've got the right idea even if your builder hasnt. You need to have air changes in the room for your own comfort and also to control the heat build up. All that equipment giving off heat before you factor in the human condition can get very uncomfortable. - especially with a pj running.
There are ways of sound proofing without comprimising on ventilation and vice versa. A domestic air con unit will only chill the ambient air, you need to have a constant turnover of fresh air or you'll be asleep before the film finishes.

This can be done using sound attenuating transfer boxes but you would normally build these in with the construction.
The requirements for the provision of ventilation in dwellings are detailed in the Building Control Regulations and apart from providing fresh air to breathe, are meant to restrict the build up of moisture and pollutants, which would otherwise be a hazard to health.
Ventilation rates are usually quoted as 'air changes per hour' defined as the volume of ventilation air moving through the room per hour, divided by the volume of the room itself.
BS 5449 stipulates 1.5 air changes per hour for a living room.
Have you asked your builder how he intends to achieve this?