Actually, the reverse of what you suggest is what happened. The name "Weimar Republic" (Republik von Weimar) was not used until after the Weimar Republic was effectively dead under the rule of Adolf Hitler; it was used then to refer to the then-gone government. During the period of the Weimar Republic, the name for Germany officially was Deutsches Reich, ir German Reich, where Reich means something more like "nation" or "land" in English than "Empire" as it's often translated. The popular name for Germany among Germans varied over time and depended on one's political orientation, though "Deutsches Republik" (German Republic) was apparently rather common.

The second question is rather more complicated, but it was pretty apparent that the Nazi government was not going to be the same as what had before almost right away. After being appointed chancellor, Hitler proceeded pretty much immediately to crack down heavily on opposition parties and supporters. Only a few months later, the Reichstag fire (February 27, 1933) created a sense of national emergency, and the Enabling Act passed the next month (March 27, 1933) gave the chancellor (Hitler) essentially free power to govern without having to deal with the Reichstag or Reichsrat at all. Some people might argue that the Weimar Republic died on the day Hitler became chancellor, but there was no doubt in anyone's mind that it was over once the Enabling Act was in force.