I'll try to address these point by point.

A) Prussia started as a region stretching along much of the southeastern coast of the Baltic. The natives of this region were referred to as "Old Prussians" and were the dominant social and political force in the area until the Baltic Crusades. In 1466 the Second Peace of Thorn saw the transfer of western Prussia from the Teutonic Order to the Kingdom of Poland. In 1525, with pressure from the Protestant Reformation making the collapse of the Teutonic Order imminent, the Grand Master Duke Albert of Brandenburg-Ansbach secularized the Order and established the Duchy of Prussia in the eastern part of the region. It was an overwhelmingly Protestant and majority German duchy, but Albert pledged loyalty to the Kingdom of Poland in exchange for retaining a significant amount of autonomy in internal affairs. In the early 17th century (1618 or 1623, depending on technicalities) this duchy fell in a personal union under the Electorate of Brandenburg, as Albert's line died out and his family was a cadet branch of the House of Hohenzollern which ruled Brandenburg. The two states remained under this personal union until 1701, when Frederick III elevated Prussia to a Kingdom and formally unified the two realms. This Kingdom is what most people are referring to when they say "Prussia." Prussia continued to exist as a state within Germany until 1947, when the Allies dissolved it.

B) Technically? The German Empire was established by Wilhelm I, who was King of Prussia. He, Frederick III, and Wilhelm II retained the title "King of Prussia" and Prussia was seen as the leading state in the now-unified German Empire. Interestingly enough, the German Empire functioned in many ways as a federation with Prussia as the dominant political, economic, and military state. Other constituent kingdoms such as Bavaria retained their own militaries. However, Prussia contained the majority of the Empire's population and land and was thus able to dominate its political agenda in the Bundesrat without the support of the other kingdoms. So I suppose you can say that Prussia continued to exist as a state within the unified German Empire, but in practice the German Empire was now the primary realm of the Hohenzollerns.

C) After the abolition of the Holy Roman Empire during the Napoleonic Wars, the remaining German states created the German Confederation. It quickly became dominated by the rivalry between Prussia and Austria, and as a result the Confederation was rarely effective. It collapsed in 1866 when Prussia and 21 other small North German states left to form the North German Confederation. Austria, its South German allies, and the new Confederation went to war shortly after. The Prussian-led North German Confederation won decisively. This sidelined Austria and established Prussia as the dominant power in Germany. Final unification came after the Franco-Prussian War, when the North German Confederation and the remaining independent southern German states (Bavaria, Baden, Wurttemberg, and Hesse-Darmstadt) fought France and claimed the ethincally German regions of Alsace and Lorraine. The political and nationalistic fervor resulting from the war proved to be the tipping point in German nationalism, and the four aforementioned states agreed to join with the North German Confederation in the new German Empire with the King of Prussia and former President of the North German Confederation as Kaiser.

Hope that answers your questions.