No, but there are depictions of children in war zones who are distressed. They are in the background and serve to illustrate the horrors of war, but are not a direct part of the story.
In addition to the horse deaths in the battle scene, there is also a scene in which Diana sees horses stuck in the mud being beaten repeatedly by carriage drivers.
Open to interpretation. As far as main characters go, a beloved aunt dies in battle. As far as people in the background of war goes, a whole village dies which presumably includes parents, but no children are depicted in the brief glimpses we get of it.
Definitely no. I'm not sure why people are saying yes- it takes place during WWI when nuclear weapons didn't exist. The closest thing is the Wonder Woman's arm bands give off some sort of non-lethal shock wave a time or two.