The universe is expanding and that dominates at distances of galaxy clusters and larger but at smaller distances other forces dominate. The Andromeda galaxy is heading towards us because gravity is stronger than the expansion of the universe at that distance. Just like gravity overcomes expansion in our solar system and atomic forces overcome expansion in your body.
The balloon analogy is just an analogy. The galaxies as dots aren't connected to the balloon and other forces are at work, not just the expansion.
The big bang theory describes the history of the universe from very early (hot and dense) to now, it doesn't describe how the universe got to that state to begin with.
http://map.gsfc.nasa.gov/universe/uni_age.html
There's one link on how the CMBR is used to find the age of the universe.
The wiki entry is pretty good as well
http://en.wikipedia.org/wiki/Age_of_the_universe
And the cosmology tutorial is a great resource
http://www.astro.ucla.edu/~wright/cosmolog.htm
http://www.astro.ucla.edu/~wright/cosmology_faq.html