How to find out the when we had a common ancestor with a particular animal
TimeTree [http://www.timetree.org] is one site that allows you to look at times from divergence of various taxa. Their FAQ describes how they calculate the times by pointing to this paper: Tree of Life Reveals Clock-Like Speciation and Diversification, but in general they used standard molecular clock approaches. You ask about downloading their database: they say


  The TimeTree Database data is currently not available for download. Methods for making these data available in the future are currently under discussion.