Yes- if your child's name is on the invitation, they are included in the invite. We have a large family that consists of many children and we want everyone to join in the celebration.
Whatever makes you feel comfortable! The ceremony will be in a church, so be mindful/respectful of that. If you want to bring comfy shoes for the reception- go right ahead!
If your invitation was addressed & Guest - then yes, you are able to bring a date.