Why Milky Way and Andromeda are being drawn together if there was 'Big Bang'? Scientists used to use following reasoning:  

most galaxies are red-shifted $\implies$ there was a 'Big Bang'  

Why this is being considered valid since not every galaxy has such property?
The evidence for expansion is that the redshift is proportional to distance.

The redshift of a galaxy can be divided into two components: that due to the cosmological expansion, which stretches the wavelength of light whilst it travels towards us; and a peculiar motion with respect to the cosmological expansion, which causes a straightforward doppler shift.

The former term increases with distance - this is known as Hubble's law. The redshift here is always positive (ie always a stretching of wavelength). It is this that tells us the universe is expanding. The latter term is caused by the gravitational effects of other nearby galaxies and clusters of galaxies on the galaxy in question. It is typically of order a few hundred km/s and can be positive or negative (ie it can cause a redshift or blueshift).

Redshift due to cosmological expansion completely dominates at distances greater than a few hundred million light years. Up till then (and Andromeda is only 2 million light years distant), peculiar motions can result in blueshifts for some galaxies.