In "Dover Beach" what "reality" is a subjective projection of the speaker and his own anguish? What seems to cause his melancholy? Does he find hope?

Matthew Arnold projects his anguished viewpoint of the world upon the beautiful scene from his window, and he does it primarily through the connections he makes with what he sees to the melancholy that he feels about the world.  To him, the sound of the ocean waves on the beach is not a calm, soothing, serene sound as many people often find it; rather, he thinks that it brings in "the eternal note of sadness" and the "turbid ebb and flow of human misery."  So whereas other people might connect that peaceful scene to a calming, happy feeling, Arnold connects it to sadness and misery.  This is him projecting his own worldview on the scene before him.

Also, he picks his words carefully as he describes the scene, using words that reflect his own emotional state about the world.  Note the harsh, negative word choices:  "grating roar," "fling," "tremulous," "vast edges," "naked shingles," "darkling plain," "confused alarms" and "clash".  All of these words reflect Arnold's reality of anguish and despair that he feels over the state of the world, which taints even a serene view from a window.  He feels that any beauty and goodness have disappeared from the world--in the last stanza, he goes through an entire list of things he feels are no longer in the world.  He feels that all of those things, and faith, have departed, leaving only violence, darkness, exposure and misery behind.

The only possible hope he sees is if he and his love are "true to one another."  By forging meaningful, loving and honest relationships, he feels that he can at least protect himself from the awful darkness descending on the world.  It's not much hope, but a little tiny flare of it.  I hope that those thoughts helped; good luck!