Yes, I usually pay with my app, which is the same as gift cards (I load the gift cards on my online account).
Don't know about the birthday treat (tend to think no) but possibly the Citywalk starbucks would be able to redeem it. I know the Disney Springs Starbucks are different from the parks ones. I am a little ticked with Starbucks because now you can only use your birthday treat on the actual day of your birthday (they used to give more days to use it!)