What is the dating of the earliest extant Latin manuscript of the Hebrew Tanakh (Old Testament)?
What is the dating of the earliest extant Latin manuscript of the New Testament?
Who is in current possession of these manuscripts?
The Latin Manuscripts of the New Testament are of two main families. The oldest complete New Testament in Old Latin (the version that predates Jeremone's Vulgate) is known as Codex Ardmachanus and comes from ca 850. Codex Speculum contains all of the New Testament except 3 John, Philemon, and Hebrews (housed: Saint Cross monastery (Sessorianus) in Rome). However, the oldest partial is of the Gospels and comes from 350 (Codex Vercellensis, housed: Vercelli City Library). This chart lists out all the known Old Latin manuscripts and where they are housed. I found nothing with more than pieces of the Old Testament. Mostly Genesis with the oldest (Codex Bobiensis) being 550 (housed: National Library of Naples).
The oldest known copy of Jermone's Vulgate to have the complete New Testament dates to 541 (housed: Hochschul- und Landesbibliothek Fulda). The oldest partial NT is of the Gospels and comes from 450 (housed: Bibliothèque nationale de France). This chart shows the known Vulgate manuscripts and where they are housed. Codex Theodulphianus is the oldest with the Old Testament and it comes from the 8th or 9th century (the chart lists it as tenth but the page itself says 8th or 9th. It is also housed in the Bibliothèque nationale de France).