Does chest xray of pa lateral take a picture of the lungs?

Yes: Chest x-ray pa and lateral means your doctor wants to see lung in anterior posterior position and to see heart and behind it in lateral it shows the lungs, heart , and upper part of body , neck to abdomen. It shows if there is pathology in lungs, heart or mediastinum.

Yes: Pa (posterior-->anterior) is the frontal view of the chest which includes both lungs. The lateral view is the chest seen on profile (from the side view).