Jadwiga of Poland became a King rather than Queen because by law the ruler had to be a King.

Her great uncle, King Casimir III didn't have any sons of his own so he chose his nephew Louis of Anjou (Louis the Great in Hungary) to succeed him. To ensure the Piasty dynasty, part of the agreement included that the law of succession could only be passed to Louis' future sons. However, he himself only had daughters and in 1374, convened the King's Council at Kosice in Hungary to persuade them to change ancient laws to recognize the rights of succession for women. The Council opposed the demand but after some compromise (to the benefit of Polish nobles) they allowed it.

He originally chose his daughter Catherine to be the heir but she died before him so the throne passed to Maria. But this was never realized because after Louis' death she was crowned King of Hungary, which would mean there would be no union between the two kingdoms and on top of that, Polish nobles didn't particularly like the man she was betrothed to, so they chose Jadwiga. Hopefully this answers your second question.

Sources:

Górski, Karol, V. Raczyńska, and C. Raczyńska. "The Origins of the Polish Sejm." The Slavonic and East European Review 44, no. 102 (1966): 122-38. http://www.jstor.org/stable/4205716.

Halecki, O. "A History of Poland" Chicago: Henry Regenery Company, 1966.

Much like the situation with Jadwiga (the ruler had to be a king, even if female), Georgia had "King" Tamar from 1184 to 1213. She reigned over Georgia during its golden age, and is still known in Georgian as "Tamar mep'e" (მეფე, mep'e being Georgian for king). It was during her rule that Shota Rustaveli, a legendary Georgian poet, wrote what is now considered the national epic, "The Knight in the Panther's Skin" (ვეფხისტყაოსანი, vep'khist'qaosani).

I remember hearing something about a "King Christina" of Sweden, on the same basis presumably - can anyone clarify?