Arceus's dimension does exist, and it is shown as being separate from those of Dialga, Palkia, Giratina, and Pokémon/humans. (At one point its dimension is shown overlapping the Pokémon/human dimension on the Time-Space Axis, if I remember correctly.)
As far as what's in there, the most we see in the 12th movie is Arceus resting in a massive, galaxy-sized swirling vortex of golden energy (what is supposedly its own energy, which sends shockwaves across other dimensions). In the background are other galaxies strewn about randomly. What the other galaxies are for, we'll probably never know. Other galaxies Arceus created? Doorways to other dimensions? Simply things the artists added to the background to make the dimension seem massive (and totally awesome)?
In the ending credits (which were butched by Cartoon Network, as usual), a clip of Arceus plays showing it roaming its dimension for whatever reason. Its dimension is probably just filled with whatever it wants it to be filled with. I know that's vague, but that's honestly all I can conclude - what would a Pokémon that can create entire dimensions want in its own dimension? What does it do there? Again, we'll probably never know for sure, but speculation is always fun!
