Diane Hamilton humans came out of Africa in several waves, migrating North and East over thousands of years. So yes we are all related, there is evidence that at one point there were a few as 5000 humans alive. Whether there was an original pair of humans really depends on how you define original but personally i think that it is an over simplification. This is anthropology rather than genealogy.
As for DNA proving your links to dead people your friend is sort of right, without a sample of the dead person's DNA you cannot prove a link. There are two ways that DNA studies work, one is to look at density groupings so if your DNA is XYZ and testing has only ever found XYZ in say Mongolia then you must have a Mongolian ancestor.
The way to link you to specific ancestors is to follow paper trails, if you and your cousin both have a DNA test they can look at your results and compared the similarities, knowing who your shared grandparents are experts can deduce "this bit has come from that side of your family". With long enough proven documented ancestries this type of study can push way back, the DNA studies of the Rurikid are attempting to prove the ethnicity of a man (known to us as Rurik) who lived in the mid 800s.
That's a _very_ basic explanation but should be good enough for a chat at work :)
In regards to race, the concept of race is a social construct because there is no basis to divide humans along racial lines other than because we want to.