I've been breastfeeding at the time we've made two of our WDW trips. Everyone has given you great advice - the baby care centers are wonderful, but there are lots of places throughout the parks that are wonderful, too.
Certainly BF is the most natural thing in the world, and should be supported by any thinking individual. On the other hand, being discreet is a courtesy to those around you. I agree that most people won't notice - but I wouldn't BF in someone else's photo unless I had NO choice.
My guess about the lady sitting close to the characters is that she was tired, her baby was tired, and she sat down and began to BF without realizing how close she was to the action! And once you've started, it can sometimes be very difficult to get up and move without causing a major distruption for your baby.
I also understand the need to dry your nipples in the air, but it isn't difficult to accomplish without showing them to everyone!! I believe you have a RIGHT to BF when your baby needs it, but common courtesy should not be thrown out the window.