I'm wondering if European commoners in the early nineteenth century knew who their royalty was. I'm sure not all of them would have recognized them face to face because there may have been little interaction, but would, say, a teenage girl (a commoner) know the name of the monarch running her country?
The answer is very clearly yes. Images of European monarchs were commonly reproduced on coinage, in newspapers, and in public art. Not only sovereigns but princes and minor royalty were also usually recognizable to common people. Also, most monarchs in 19th C. Europe were (or hoped to be) considered military leaders. If you're talking about "early 19th C. Europe," you're talking about the age of Napoleon. These wars were closely followed in every European country. Napoleon wasn't "real" royalty, but he was instantly recognizable and very well-known to everyone on the continent. His opponents on the field of battle, including the Prussian, Austrian and English monarchs, were feted during and after the wars to the point that their faces and names would have been widely known. Princes of smaller German states (e.g., Hesse, Mecklenburg-Schwerin, Reuss) had coins and medals minted with their likenesses on them. Numerous newspapers printed comics and illustrations. Deaths and coronations were huge, popular events, so transitions of power were well documented and known to the subjects of 19th C. European states. These people were famous. Even Queens were well known. Louise of Prussia (lived 1776-1810) had a virtual personality cult built around her, especially after her death. After the Congress of Vienna, even a teenage girl from Dresden could have told you that her shamed king was named Frederick Augustus, and could probably have picked him out of a lineup.
Not likely. The Church was the principal source if "news" then. They would have known about the actions of the priests, bishops, cardinals and the Pope... but rarely was there ant goid reason to involve illiterate commoners in secular government. What would have been the point?