
Zachariah S. answered 01/22/21
Interdisciplinary researcher in Zoology & Geology!
Hello! I think your question poses an interesting view, and I'll try to answer it to the best of my ability.
First, let's review your initial and first questions: what is an Earth without mountains? From this, we have to remember that mountains are a byproduct (i.e. the result) of tectonism (or the movement of tectonic plates). This is because we have tectonic plates moving around, and when two (or more) plates squish together (this is referred to as a convergent plate boundary) the rock material has to go somewhere. So as the plates squish the material kind of "piles up". Some of Earth's crust (this is also the tectonic plates) gets squished deeper into the crust and some of the crust goes up. Often times, this crust material piling up is called "a mountain building" event. I'm sure there are many helpful videos on YouTube about convergent and other plate boundaries that might be able to explain with 3D examples.
So, with all that said, an Earth with no mountains would (at least from my prospective) mean that we have an Earth out activity moving tectonic plates. This would be hard to stop, because it is the convection of molten materials in the mantle that helps drive a lot of plate movements, especially in the subduction of oceanic plates. So it would be hard to not have a moving crust, because moving molten materials inside. If the Core of the earth becomes cold, then sure we can say the process of tectonic plates would likely slow down or halt completely.
Now, if mountains were taken away completely, and we were left with a "flat earth", then we would technically the crust actually bounce back a bit. Similarly, to how if you sit on a trampoline and the trampoline goes down, but if you remove yourself from the trampoline, the trampoline will try to go back to the original state of flatness. So removing the weight of the mountains will cause the mantle layer to want to "bounce back" and will move the crust up/down until it is back in equilibrium again. This same process of crustal equilibrium occurs with heavy ice glacial melted since the last ice age, creating large depressions in earth's crust because of their weight.
For reference, I went over questions about the weight of mountains and tectonic plate movements in a university senior/graduate level Solid Earth Geophysics course. Hope this is somewhat helpful. I would welcome other answers in case I missed something.