How do I track the history of changes to a given verse's actual text? How do I discover it's "lineage"? Has there been a project that traces the genealogy of each verse of the scripture?

I.e., for each verse, it traces back through all the documents it was translated from, the default translations at the time, ... all the way back to historically verifiable fragments?

I'm not talking about a generic "this book of the bible came from XYZ". I'm talking about details at the grainularity of a single verse. I.e, for each verse, trace it's lineage, through all the translations, back to the original inspired writers.

Thanks!
The type of work you are looking for is called an apparatus, such as Bruce Metzger's Textual Commentary on the Greek New Testament which explains the choices made in compiling the UBS4 Nestle-Aland 27 (one of the best Greek texts, pulling together most of the known codexes, manuscripts, and papyri out here)

When using an apparatus, you should become familiar with the distinction between the texts (Say, codex Sinaiticus, an aleph, vs. Codex Vaticanus, a B, what a papyri is, what an Mss. Is, etc.) This link, a guide to the critical apparatus of the BHS (Biblica Hebraica Stugartensia) does a pretty good job of explaining the approach that scholars use in determining the lineage of a text.

It will talk about what manuscripts inform the current text out there, and it will discuss why a particular variant should be chosen. Additionally, it grades the certainty with which the scholars believe it to be "original". 
This link is not comprehensive, but gives an excellent overview of some of the more problemmatic variants. An example variant would be described as such:


  2 John 12:
  
  TEXT: "so that our joy may be made full"
  
  EVIDENCE: S K L P Psi 104 614 630 945 2495 Byz Lect syr(ph,h)
  TRANSLATIONS: KJV RSV NASVn NIV NEB TEV
  RANK: C
  
  NOTES: "so that your joy may be made full"
  
  EVIDENCE: A B 33 81 1739 1881 lat vg cop(north)
  TRANSLATIONS: ASV NASV
  
  OTHER: "so that my joy may be made full"
  
  EVIDENCE: cop(south)
  
  COMMENTS: In later Greek the words for "our" and "plyour" were pronounced alike. It is likely that copyists misheard "our" as "your," influenced by the two previous occurrences of "you" in the verse.


In this instance, particular attention should be paid to the "Evidence" entry.  Each letter or abbreviation refers to a particular manuscript that contains the variant. 

Finally, if you check out the Net Bible, their public domain text has incorporated the apparatus into the footnotes in most places where there is any question. For example:


  3 tc ‡ Most witnesses ([א] A P 1739 Ï sy) have καί (kai, “also”) before the article τόν (ton). But the external evidence for the shorter reading is significant (B Ψ 048vid 33 pc sa), and the conjunction looks to be a motivated reading in which scribes emulated the wording of 4:21 (ἀγαπᾷ καὶ τόν, agapa kai ton). NA27 places the conjunction in brackets, indicating doubts as to its authenticity.


Again, notice the 'most witnesses' and the list of manuscripts.  The 'A' representing Tischendorf's stolen Codex Sinaiticus is properly called א in this work.