How (if at all) did the religious and ethnic demography of the Levent change as a result of the Crusades? Did any groups lastingly convert to Christianity? Did any sizeable group of Europeans stay and leave genetic traces?

According to Fr. Sidney Griffith, the major legacy of the Crusades on the demography of the Crusader states was the extended destruction of native Christian communities in much of the Levant. His work, The Church in the Shadow of the Mosque is a breakdown of the history of Christian communities in the Islamic Middle East, from the period of conquest up through to the Mongol conquests that rolled the Middle East in the 13-14th centuries. That is not to say that these were the only changes that resulted, and I'll get to some of the other ones as well.

In short his book lays out that Christian communities, broadly divided into a few categories, such as Jacobites (a group of eastern Christians that I will talk more about in a bit), Melkites (broadly speaking Greek Orthodox Christians), and a smattering of other "Eastern" churches such as the Syriacs (also called Assyrians and whom Griffith refers to as Arameans for their liturgical language) and Armenians, all were in a period of sustained decline from the period of the Islamic conquest. There were other denominations present as well such as Georgians, Copts, and others, but they were a very small portion of the population as a whole. This slow decline ebbed and flowed at times, increasing in pace in certain situations, slowing for others, but was inexorable over time. This was due to a variety of factors, conversion to Islam chief among them as it became increasingly politically, and economically, advantageous to adopt the Islamic faith.

This process of slow decline was radically altered by the arrival of Latin Christians in the Holy Land as a result of the success of the First Crusade. The relationship between the crusaders and the native Christians of the Holy Land was not straightforward and was quite tumultuous. Crusaders at times married into and allied with native Christian societies, especially the Armenians, but at other times were responsible for horrific atrocities aimed at them. (This varying treatment has been termed as 'rough tolerance') However the collapse of the Crusader states and the eventual triumph of Islam in the region, and the eventual ascendancy of first the Mamluks and later the Ottomans brought about a collapse in the Christian populations of the Holy Land. Whereas the decline under the centuries after conquest had been gradual, following the end of the Crusades it was precipitous. Griffith connects this to a shift in Isalmic thinking on the status of dhimmi populations following the Crusades and conscious effort to expel Christian communities from many areas dominated by Muslims.

Other Christian communities were lost during the tumultuous end of the Crusader states and the influx of Mongol, Turkic, and Egyptian armies/states that rolled through the Middle East in the 13-16th centuries. Associated with the harsh rule of the Latins, and the horrific sacks of Jerusalem and other cities, the Christian communities were mistrusted and targeted with both sporadic and systematic violence at different times.

However these were not the only changes that occurred. The arrival of Latin Christians in the East created new opportunities as well as for Christian communities in the east and some denominations in the Levant eventually took the chance to move closer to Rome in their theological bent. Indeed a few denominations moved back into communion with Rome, among them were the Jacobites/Maronites that I mentioned earlier. Later on some other churches did so as well. This created new connections between the Latin and Eastern Catholic churches that have continued down to modern times. A full recounting of the histories between these churches is a bit out of my scope though.

As for genetic studies, the simple and short answer is no.