Hi profilactic,
It does sound as if this girl is interested. All those questions and how do you handle them? Well it depends. Some say you should be confident and proud of yourself and answer them directly and others say to make up a funny answer or just ignore the questions.
I have literally just finished watching a Lance Mason clip where this situation is answered well. It depends on what other interaction has taken place and how you have conveyed yourself in that. If for example you have been talking a while and in general have come across as funny it's probably wise not to answer her questions jokingly. She may think you can take nothing seriously and you're typecast as a joker. The opposite may also apply. Girls don't want one dimensional men.
Try this link. About half way down the page there are 2 long video clips and Lance gives suggestions on how to answer these sort of questions, plus a lot more besides
http://www.pickup101.com/birthday/
http://www.pickup101.com/birthday/