Why won't people call them Dwemer instead of Dwarves? is that some kind of funny as High elves are to Altmer and Wood elves are to Bosmer? Dwemers context is Deep elves no? they were only Dwarves among Giants.. blast you to hell Oblivion
As people have already pointed out(but I'll be more to the point) Dwarves didn't use to mean short. In mythology, they were just elves that lived underground, which is essentially what TES dwarves are.
Also keep in mind, the Developers didn't come up with Dwemer first and then say, hey... we can call these guys Dwarves too. They took the concept of Dwarves, and repurposed it, and then to give them a lore correct elvish name, they called the Dwemer. So for the developers, they are essentially Dwarves and I am sure that is what they are to most common folk in Tamriel.
House Dwemer is specifically mentioned in Morrowind many times. They as well as House Dagoth where one of the houses of elves that later became Dunmer. At last that's what all Morrowind books say.
Yep, me bad, it's kind of hard to remember sometimes that they all "morphed"
But it was the same people, just looked differently before.
As far as I know, the Dwemer were an altogether different sect and race of elves from the Chimer. They did not follow Saint Veloth to Morrowind. Yes, as far as the Chimer were concerned, they were essentially another "House" but they were nearly entirely separated from the culture and ways of the House Chimer of Morrowind. BTW, they did not become Dunmer at any point, they were gone by the time of those events but in any case, not being of the same race as the Chimer, they would not have become Dunmer. Also, the Chimer did not disappear with them, which also suggest that they were different enough to be consider a separate race.
So, in conclusion:
Dwemer ceased to exist after the became the skin of their Robot God.
Chimer(people who followed Veloth to Morrowind) became the Dunmer.
They both had golden skin, because golden skin is the natural color for Mer, especially the closer you get to the original elves.