In a kingdom by the sea, (line 2)
The sea is everywhere in this poem. It comes up so many times that you might stop noticing it, but it's an important image, and it keeps this fantasy grounded in something you can understand. Like with everything else in this poem, Poe doesn't go out of his way to describe the sea. He doesn't call it "bright" or "stormy" or anything like that. We don't know anything about its color, but somehow we can feel it out there, a constant presence. The poem is about love and death and angels and all the rest, but it's anchored in the natural world.
A wind blew out of a cloud, chilling (line 15)
That natural world can be scary too. Here we find out that the thing that killed Annabel was a wind from a cloud. Our speaker's theory is that it was sent by the angels, but it's the natural world that does the dirty work. Again, we're not talking about happy nature poetry here. Poe's world, natural or not, is always a little dark, scary, and ugly, and even slightly insane. When even the clouds are trying to kill you, you know you're in trouble.
For the moon never beams, without bringing me dreams Of the beautiful Annabel Lee; (lines 34-35)
Could this be a comforting nature moment? Are things OK now, even for just a minute? Maybe, since Annabel is the one person he misses most, and the moon brings a reminder of her. At the same time, we don't know if these are happy dreams, and it seems like they might not be, since our poor speaker always seems so tormented. He doesn't want to be separated from her, but maybe he should be. He seems like a slave to the memory of a dead girl, and the moon seems to be helping to keep him that way.