When we went it was during a cold spell so the giraffes had to remain in their "barn". The tour took us inside the barn and we got to see the giraffes in there. We were so close that we could have probably touched them, but that's not allowed. I would assume that when they're outside, you may or may not get really close. The truck drives around the safaris and the animals are free to roam.
Regarding the 13 month old, the safari isn't really geared for infants. Once the safari part starts, there is no chance to go to the bathroom or take the child somewhere else if they get fussy. You must remain in the truck and there are no seat belts, so you'll have to hold him. The safari is about 90 minutes long. After that, you head into AKL and down to Jiko. Dinner is a three course family style meal with your fellow safari mates. It takes another 90 to 120 minutes.