In short: No.

Longer answer: No, but they did.

Even longer answer:

The situation of the German principalities that survived mediatisation and the Reichsdeputationshauptschluss (such a lovely word) didn't really change much. While the larger of them were now kings, or in the odd case of Hesse - Electors (don't ask), they were still firmly wedged between three of the European superpowers, Prussia, Austria and France. While the first two were fighting for political supremacy in the newly-founded German Confederation, a loose coalition of states, the French still had their eye on the Rhine.

This pretty much meant that the smaller principalities were used like playthings by the larger states, clinging to their overlord and hoping that everything will work out. Outside of revolutionary overtures that had to be quelled in '48/'49 it sort of did, but only for Prussia, when it annexed most of the German states opposed to it in the Austro-Prussian-War (Hannover, Frankfurt, Hesse-Nassau, Schleswig-Holstein, Hesse, part of the other Hesse, lots of Hesse, really) and formed the North German Federation with its allies. The only properly independent princes left at this point were the Grand Duke of Baden, the King of Württemberg and the King of Bavaria. Austria pretty much was left to sulk and do nothing in regard to Germany from this point on - Prussia simply won.

Cut to the Franco-Prussian-War. The German states unify to form the German Empire, which notably was still a federal state. This meant that even though Prussia was by far the largest state of the German Empire and dominated it politically, every single itty-bitty formerly independent principality (all those tiny Saxon duchys!) or free city still had its own government, legislature and constitution in addition to the federal one. With the transition to the Weimar Republic, that didn't change much - except that every single prince and king was ousted, of course. They kept their wealth and land, but had no political influence written into the system itself, only soft power.

I'll keep it short in regard to today because of the 20 year rule and firmly say no, they do not. Most Germans today consider nobility as a curiosity of the past that has no bearing on everyday life or politics. They'll have their fancy events and sometimes one of them will pop up in politics (looking at you, Gutti!), but it's not institutionalised in any shape or form.