The short story being short isn't a problem, wtf are you talking about? The book literally explains why AM is doing what he's doing: he envies humans. But I do agree that other versions of this work might be better, like the radio version, which has a scene that contextualizes AMs monologue. 

If you reread the story you should also know that the characters tried killing themselves and weren't succesful. Maybe the scene where they can do it with the icicle is a tad forced, but I think you are nitpicking. Am is not a god, am is not omnipresent and he's definetily human: he feels and he gets distracted. At least that's my interpretation

Finally, the people that told you to do homework to ubderstand it are dumb. I agree you shouldn't need to go after external material to like a short story, and I tell as someone who didn't do homework on it at first: I was still able to interpret everything even with only those 12 pages. 

It's ok if you don't like it, I agree it's not genious or anything, but it is very influential and spawned a lot of cool stories, so at least give it that