If your EGR was malfunctioning you would get a check engine light. There are sensors to detect such things.

EGR is not used to help warm up your car. It is not used when the engine is cold. Its only used once the engine is warm and at partial throttle. As Pickles said, it does close at wide open throttle (WOT).

Air going in/out of the oil fill is normal when the cap is removed.

These are all really odd questions. Why do you think you have a problem? It would easier to diagnose if you describe your problem you're having rather than saying what you think might be wrong.