Is it safe and relatively doable for a single mom and two kids to rent a car and drive from Pamplona to Barcelona?
Sure, it's safe and do-able. Four-lane divided highway all the way, altho much of it is a toll highway. Driving time would be about five hours. However, driving IN Barcelona is pretty crazy. Depending on when you want to go, you can get advance tickets for about 24 Euro pp and they'll get you into Barcelona in about 4 hours.