If the invitation says "And Guest" then yes, if not, then no we would prefer it was just you and whomever is listed on the actual invitation.
No, this is an adult reception only.
The ceremony and reception are at the same place so you will only have to park once. The venue is located in the Warehouse district in New Orleans. There are plenty of parking lots within a couple blocks where you can pay to park and street parking is also available and free after 7 pm.