Beowulf

We are beginning our hero unit in English, and had to respond to some questions before we started our first story. Now that we have completed the first piece of literature, I am still thinking about the initial questions: what qualities do I think a 'hero' must possess, does a hero have to overcome a major feat in order to be considered great, and what is a 'hero'? At first I thought these questions were fairly easy to answer, but after reading the story Beowulf, I will admit I am a little confused. The character of Beowulf contradicts many aspects and simply does not live up to in some ways, what I believe a hero is.
I said that a hero must possess a great inner strength that propels him/her to fight for what is right. A hero must also be fair, caring, determined, and loyal. I do not think that a hero necessarily has to overcome so huge physical or mental challenge, such as slaying a dragon or solving a Rubik's cube (I have no idea where what came from, but to me it's a feat). Because, what if the guy trying to save the girl from the beast doesn't succeed? And the child prodigy simply can not solve the puzzle? What if your grandfather can not fight off the cancer after a 2 year battle? To me, all of these people are heroes, even though they did not conquer what they set out to do. The fact that they took the time to fight for something, and give it everything they had makes them a hero to me. Here, winning doesn't matter as much. A hero must be able to recognize the essential parts of himself and of the world, the strengths and weakness of both, in order to perform at the elevated level that we see them as.
As a girl who used to watch Disney movies every day (mainly the ones with the princesses, my how things have changed) the description above is how I picture a hero. However, after reading Beowulf, I have become skeptical. I really enjoyed the poem as a whole, but I can not see how Beowulf is an epic hero, or any kind of hero for that matter. In m opinion, he does not fit my criteria for a hero at all. I found Beowulf, although helpful towards the Danes, arrogant and annoying. He knows how wonderful he is and he knows that everyone loves him and it got on my nerves how he always had a remark to everything anyone said about him. When Unferth tried to put him in his place, Beowulf retorted with a "Well actually, THIS is how the fight went down (insert exaggerations here)...it wasn't a big deal." And when the king of the Geats died and his son passed away as well, it just MADE SENSE to give the kingdom to Beowulf. I mean, come on, does everything have to go right for this guy? He didn't stumble once. Nothing was difficult for Beowulf, even dieing was pretty easy for him. And Beowulf was not made into a personal character, he was created more as an ideal, I believe, which I think makes it hard for the reader to relate to. We don't know if he cared about anyone, or even himself. He reminded me of Achilles in a way, who never really fought for anyone in the Trojan War other than himself. Both men did what they were expected to do and then got out of the way. How can one be determined about something if they have no connection to it? It's impossible. Beowulf never sacrificed anything; he never knew what it felt like, what it meant, to lose until the end. And people today talk about him as a hero. He is classified as an epic hero in my English notebook. But I simply can not agree...I don't see it.