No, "enter" here is from "enter and leave the cemetery", so it means they enter cemeteries, while having this feeling of respect at the same time.

Many who visit any of the vast, grave marker-laden cemeteries of the western front enter the cemeteries feeling a deep sense of respect . . . When they enter the cemeteries they already feel respect, but when they leave, they feel, "a profound sense of waste and folly of war."

Not exactly. "enter feeling" here means "they enter (into a cemetery) with a feeling of deep respect (for the millions of dead)". It implies they are already feeling a sense of respect before, or at the moment of entering. "Start feeling" implies they will eventually have a feeling of respect at some point in the cemetery, but it doesn't say when.

It means that when they feel respectful when they enter the graveyard, whereas when they leave they feel that war is terrible. Their feelings change from the start of the experience to the end. Another example would be: "He entered the room feeling confident, but after the presentation felt like a failure". It juxtaposes how the person is feeling at one point in time versus another.