As /u/DuxBelisarius points out, the first shots of World War II were fired by the German warship Schleswig-Holstein at a Polish base at Westerplatte. However, there had been armed skirmishes on the Polish-German border that summer that killed or wounded several on both sides.

As to the last casualty: Private Kinshichi Kozuka served under Lt. Hiroo Onoda and fought with him in a private guerrilla war until he was killed in a shootout with Philippine police on October 19, 1972.

The first military casualties of WWI were on the Franco-German border, in a skirmish between French and German troops. The last military casualty, to my knowledge, was George Price, a Canadian soldier killed outside Mons 2 minutes before the Armistice.

The first military casualty of WWII was a Polish soldier of the Westerplatte garrison, near Danzig. As to the last one, I do not know.

How are you defining casualty? Immediate death due to enemy fire? Eventual resultant death from enemy action? Direct fire or indirect or explosives?