The air bled from the compressor is not cooled, it is a small tapping from a particular compressor stage or stages.
The bearings are pressurised with bleed air to assist the seals in keeping the oil/air mist in the bearing cavity.
The oil breather is there to vent the bearing cavity otherwise the compressor air would force the oil/air mist back through the labryrinth seals and into the engine. (where it could contaminate the cabin air).This breather air is dumped overboard.