Map of Arda...

[The Might]

I personally am not sure of how accurate these maps really are...
As the EoA says directly under the map:

The map of Arda shown here is based on Maps IV and (especially) V reproduced by Christopher Tolkien in volume IV of The History of Middle-earth. These maps belong to an early phase of Tolkien's mythology, and in a few cases the nomenclature of the originals has been updated [...]
It's important to remember that the original maps on which this one is based predate the stories of the Third Age, and aren't entirely compatible with the later geography of Middle-earth. In particular, the relatively small northern continent would probably need to be rather larger than is shown here to accommodate all the lands that Tolkien later devised.

So, I doubt that we should consider these maps as correct and true to the story.

As for the Dwarves. The quote you gave comes from the Silmarillion and talks about Aule putting the Fathers of the Dwarves to sleep in "far sundered places"

The same idea is again mentioned in The Letters:

And he commanded Aule to lay the fathers of the Dwarves severally in deep places, each with his mate, save Dúrin the eldest who had none. There they should sleep long, until Ilúvatar bade them awake. ~ Letter #212

And even later on in their history there was a difference between them as far as the area they lived in is considered.
We know the Longbeards dwelt in Khazad-dum.
The Firebeards in Nogrod.
The Broadbeams in Belegost.
The Ironfists in the mountains in the east.
The Stiffbeards as well.
The Blacklocks were paired with the Stiffbeards, just like the Broadbeams with the Firebeards, which probably means that they lived in about the same area.
Finally the Stonefoots, were also said to dwell in the east.

There seems to be a pretty clear difference between those in the west and east.