As far as I know the only way to "become" member of Judaism is to be born as one - and that's main question that bother me - how can one "nation" (or better to say a larger number of people) "accept" Judaism? What is the history and theology behind this conversion of Khazars somewhere in 8th/9th century?
Also, I have read there is theory that a lot of Jews today actually descended from those converted Khazars - is it true or not? What are the arguments?
Not to discourage further discussion, but I addressed the Khazar myth of Jewish origins in this comment from about a year ago.
TL;DR: Evidence supporting the existence of a Jewish kingdom in the Caucasus region in the early Middle Ages is sketchy at the absolute best, and there is no evidence at all of a genetic link between modern Ashkenazim and any known Turkic population.
Minor note: The above is absolutely untrue and people can and do convert. It was more common prior to the middle ages, when it became discouraged to seek converts due to the fact that this would put converts at risk, but there have been conversion procedures and rituals for quite some time.
Minor note: You can and people do convert, and this has been possible for a very long time. It was not as common and discouraged in the middle ages(where there was concern that converts would be endangering themselves) but it can and does happen and has been happening for quite some time.