Is it harder for people with emphysema to get enough oxygen while they're under general anesthesia?

Not usually. Room air contains only 21% oxygen. When a patient is under general anesthesia, we can increase the amount of oxygen breathed to 100% if necessary. So even with emphysema, it is rare to have problems receiving enough oxygen. However, patients with emphysema will have a harder time breathing on their own and recovering after surgery and general anesthesia.
No. The emphysematous patient will actually get more oxygen under anesthesia than they usually do. The problem comes on the postoperative period. This type of patient is very sensitive to any agents that depress their ventilation, this means most anesthetics and all pain medications. For this reason this type of patient will be watched closely on the pacu.
Sometimes. COPD results in poorer lung function. While under general anesthesia, a patient typically receives adequate oxygen, albeit less than a healthy athlete. The real challenge occurs during emergence, these patients typically require more supplementary oxygen, at least initially. Severe, critical COPD may be a containdication for general anesthesia.