What is the proper role of a king in Beowulf's Germanic culture? What does he have to do to keep his people satisfied? It would be useful to examine Hrothgar's sermon to Beowulf. Also, what responsibilities does a thane have to his lord or king?

A king is a protector for his people. He is chosen by his people as a man who is strong, reliable, powerful, trustworthy, etc. A king is someone people can trust to defend them, to conquer new territories for them, to provide them with everything they might need to live comfortably, and to rule them fairly. In order to do that, he needs to be strong enough and have a powerful army beside him.

A thane has to be loyal to his lord or king. He shouldn't disobey him, he shouldn't betray him or do him any harm. He should follow his orders, be by his side when necessary, and provide advice and/or people to fight for him. Beowulf, for example, is a thane to King Hrothgar, and he came to his aid when Grendel was killing his men.