How is original sin transmitted? Assuming the doctrine of original sin is true, how exactly is original sin passed down to the rest of mankind? Was the nature of man changed? If so, what exempted Christ from sin but allowed him to remain 100% man? 

Is there a genetic inheritance of some strand of sin DNA?

What does the Bible say about this? What does Church history have to say?
I think that the best answer here is that the Scriptures are not entirely clear on how original sin is transmitted.  One thing that we do know (or believe the scriptures to teach) is:

The sin nature is passed on to us from our fathers and not our mothers 


Sin entered the world through Adam - Romans 5:12 - not Eve who sinned first.
More directly taught in Heb 7:9-10
Christ Who did not have an earthly father did not have original sin passed to Him.


That is about all that we know (and even this view is heavily disputed.)  

Other good questions:
Is it genetic (I doubt it) or spiritual (much more likely)?  What would happen if we used two women's DNA to clone a new person, if the new creation had a soul would it be sinless?