Moses married Tzipporah, a Midianite woman (the daughter of Jethro, a Midianite priest). Before the time of Moses, it's very hard to call someone Jewish because tradition holds that Jewish law came into existence when Moses went up Mount Siani. So, Tzipporah is one obvious candidate for the first women to convert.
Another early candidate would be Rehab, the harlot of Jericho. Rehab helped Joshua spy out the town, and post-biblical Jewish tradition holds that Rehab converted and married Joshua after the fall of Jericho.
But I wonder if you're thinking of Ruth, who was almost certainly not the first woman to convert, but who is viewed as the prototypical female convert.
According to Jewish law, a Kohen can't marry a convert to Judaism.
In order for a person to be a Jew they have to either be born to a Jewish woman or convert to Judaism following halacha (Jewish law).
Lots of reasons. One is that Judaism doesn't believe in encouraging people to convert. If someone wants to convert to Judaism, they're welcome to (although discouraged at first), but Judaism has no mass programs to get people to convert. Nor did it usually have forced conversions like the other religions did. Another reason is that Jews have been heavily persecuted over the years and many were killed.
He didn't.
Nothing. Jews do not seek converts. If a person wishes to convert to Judaism, they are permitted to do so, but Judaism teaches that Judaism should not be proselytized.
No, Obama is a Christian.
yes
Yes, according to tradition.
Only Judaism
His father is a convert to Judaism and his mother is Jewish.
A new Jew.
Yes, but Judaism says that the faith of the child depends on the mother. If she's not Jewish, but wants her child raised Jewish, he or she has to convert after birth.