Jewish communities in the Rhineland are very old - the community in Worms (Borbetomagus) is said to have been founded in the 5th century BC, i.e. during the first diaspora. During Roman times, Jewish communities were established in many larger towns. The first written evidence on the Cologne community is a regulation issued by Constantine the Great in the early 4th century.
In the Holy Roman Empire, Jews were outside traditional "tribal laws" (Saxon, Frankish, etc.), and regulated by foreigner laws (applying to all merchants from "abroad", irrespectively of their religion) instead. This implied that Jews did not pay their taxes to local nobility or the cities, but directly to the Emperor. Consequently, during the high middle age (10th-13th century), several emperors, as well as territorial bishoprics (Trier, Mainz, Cologne) that also acted as Imperial institutions, actively encouraged Jewish settlement in order to strengthen their tax base, in the case of bishoprics also to counteract the attempt of cities to emancipate from bishop control. They appear to have in particular targeted Jewish communities from areas threatened by the Arab expansion, i.e. Sicily and Southern Italy.
As to the "Abruzzi connection", it might be worthwhile to investigate the fate of the citizens of Lucera: Frederick II in 1224 re-established the city as a "refuge" for Saracenes from Sicily, who were granted religious freedom there. In 1300, Charles II of Naples sacked the city and massacred or exiled most of its inhabitants. I could imagine that the promise of religious freedom did not only attract Muslims, but also Jews to Lucera. After the prosecutions during the first Crusade, the situation of Central European Jews had stabilised - a new wave of pogroms only set in around 1350. So in 1300 the Rhineland might have appeared a safe place to go for Lucera Jews.
On the Khazars: Georgia and Armenia had a sizeable Jewish population, and several Jewish communities there claim to have been established in the 6th/5th century BC, i.e. during the first diaspora. When East Georgia was under Persian rule in the 17th and 18th century, several thousand Jews were forcefully relocated to Iran. As such, genetic similarity between Georgian and Persian Jews should not come as a surprise, and is no argument against a "Khazar connection."
http://en.wikipedia.org/wiki/Georgian_Jews
http://en.shater-izhaka.ru/judaism/georgia/
The economic base of the Khazar empire was intermediate trade between the Arab Kalifates and the Baltic Sea region. Baghdad, the Khazar capital of Ity (Astrakhan) and the Varangian trade hub of Staraya Ladoga were more or less established simultaneously between 750 and 760 AD. Soon after, massive inflow or Arab Denars into the Baltic Sea region commenced that lasted until approximately 990 AD, attested by various coin finds e.g. in Lübeck. It is reasonable to assume that Jews played a major role in this trade network - for their traditional role as merchants, and as Judaism had become the official religion of the Khazar elite. Such a role should have re-inforced (genetic) links to the Jewish community in Bagdad, and also have spread Jewish communities along the main trade routes.
http://en.wikipedia.org/wiki/The_Trade_Route_from_the_Varangians_to_the_Greeks