Excavating the Mines of Moria
 

The Mines of Moria were vast, maybe as much as 40 miles across, with huge habitable areas on many levels towards the eastern side where the main gates were, and the "working" and mining areas towards the west side.

The Dwarves' excavation of Khazad-Dum must have been an incredible undertaking over the ages, but one question - where did they put all the rock and rubble they dug out of the ground ?

Hm. Possibly they just piled it up in the areas near the Gates, making small, artificial hills that over time came to look natural? The only other option I see would have been to haul it away. But that seems a rather pointless (and tedious) undertaking.

They may have also have sold it off the help build walls and towns and things... as well as statues. also the misty mountains probably have some pretty good areas to deposit such things

Slag heap
 

Made me think of the gravelly-stony scree slope that Bilbo, Gandalf and the Dwarves slid down in their escape from the Misty Mountains. Though too far North for a Moria slag heap.

Up the Valleys they've landscaped a lot of the old slag-heaps and they look reasonably tidy now after only 50 or so years, so I guess after a few thousand years of weathering in Middle Earth it would be difficut to tell what was mine-spoil and what was natural.

The dwarves may have thrown a lot of their waste into pits and things inside the mountains as well.

...explaining just how they managed to wake certain things up...

It's also possible that the mines began as part of a large natural system of caverns, and thus would have had much less unwanted material to dispose of. Though Dwarves considering almost any kind of rock or metal "unwanted" seems a bit out of character. I would suspect that they traded them with Elves and Men who wanted building materials for stone structures or monuments or such for foods they could not obtain or grow underground. One doesn't hear a lot about Dwarven farmers, after all. :)

I think that's a good part of an answer. Another, depending on the
quality of stone and other material, sold or given to Gondorians for their many
building projects in Gondor and Calenardhon, if not as facing, then
as filler. Btw, why not employ dwarves as builders/co-builders of
some of them---think Gimli after the War of the Ring in Minas Tirith
and the Caves of Aglarond.

It is recorded that the drawarves in the ealier ages were the most important builders of roads im Middle-Earth. Think about old Dwarven-Road running from the Havens of the Falas east at least as far as the eastern edge of Mirkwood and proberbly beyond that further on to the eastern Dwarven kingdoms. Road building need a lot of stone, especially if it is a good road and these were good roads considering how long they were in use.

Respectfully
Findegil

JRRT does at one point say that Durin discovered the caves above Dimrill-dale and chose to make his home there.

In Chapter 10 of The Silmarillion, OF THE SINDAR, it states that, Far to the east were the most ancient dwellings of the Naugrim, but they had delved for themselves great halls and mansions, after the manner of their kind, in the eastern end of Ered Luin. These were Nogrod and Belegost. It goes on to say that greatest of all the mansions of the dwarves was Khazad-dum. The Sindarin name is Hadhodrond which translates as Khazad(dwarf)-hall (rond can be vaulted or arched roof, and dome), also known as the Dwarrowdelf. In the index to The Silmarillion, Christopher includes this:- See Khazad; dum is probably a plurural or collective, meaning 'excavations, halls, mansions.