Going off of my other question here, I guessed that a cube planet could work out. BUT, would it be possible for the planet to exist and function as a (relatively) normal planet with life on it without having tectonic plates? I know that tectonic plates with a cube don't really work out too well in life's favor.
Artificial planet made by Ancients is your only option.
If your planet was "nature made", without plate tectonics and with crust thick enough to support your gigantic mountains, such planet would have very little mantle (melted core). But turbulence of such core creates magnetic field protecting the atmosphere.
Mars has no atmosphere because its atmosphere was swiped away by solar wind: Mars also lacks strong enough magnetosphere (protective magnetic shield) - which is direct consequence of its cold core (which prevents mountains from sinking into the mantle).
So Ancients need to take care for magnetic field - not only to protect atmosphere from solar wind, but also protect life from radiation from space.
Yes, such planet is very carefully engineered project - most likely PhD in planet engineering from advanced race.
Some interesting reading about why mountains on Earth have theoretical limit on height of about 49K feet (15km). Another unexpected (for me) force grinding mountains down: glaciers eroding valleys.
The cube would have to be artificial and relatively recent. It can't be created by natural process and eventually tidal forces will break it apart. As such any life on it would have to be either created artificially or imported from elsewhere. You can simply assume that the planet was created or given maintenance recently enough that the planet still works as planned.
Forget about cube planets, by now, and go for a real planet: Mars.
Mars has no tectonic plates, but for it being habitable it just needs some more atmosphere and heat.
In fact, if Mars were where Earth is, it would surely have basic life (at least) on its own, since polar caps would be smaller and there would be more water on them. A lake is enough to create life, from where bacteria would have colonized the planet's soil.