Absolute zero is the point at which the fundamental particles of nature have minimal vibrational motion, retaining only quantum mechanical, zero-point energy-induced particle motion.
At absolute zero would electrons move around an atom at the same speed as room temperature? Does a free electron moving through empty space have a minimal velocity at any temperature?
In the standard model of particle physics the electron is an elementary point particle with mass. As it has mass , it has a center of mass, i.e. a frame where the momentum is zero, and therefore its velocity is zero. One can always use Lorentz transformations to go to the center of mass of a massive particle. Temperature is a thermodynamic variable and has no meaning for individual elementary particles.
In addition , atoms and electrons are described with quantum mechanics, and the solutions are in the center of mass of the atom, and the electron does not move around the atom, but its probable position can be described by orbitals, not orbits, so velocity of electrons has not meaning when discussing atoms, no matter what the temperature is.
Your question calls forth a lot of history.
A long time ago, physicists knew that Maxwell's Equations provided a complete description of electrical forces. Electric charges exerted these forces on each other, and Maxwell's Equations provided a complete description of how it happened.
It made sense that electrons should orbit positively charged massive things. Maxwell's Equations described the forces that would make that happen.
However, Maxwell's Equations also showed that an electron that got accelerated would always radiate. This radiation on average would do no work. It would push charges sideways first in one direction and then in the other. But still it meant that the accelerated electron must inevitably lose energy. When it loses energy it must inevitably fall into a closer orbit, and then as it continues to accelerate it must lose more energy.
Chemists had noticed that atoms have electrons that fit into patterns. When an atom has more electrons than the pattern calls for, it tends to lose electrons and wander around with an electric charge. When it has less atoms than the pattern says, it tends to gain electrons and wander around with a charge. Generally pairs of electrons, or groups of eight or whatever, stay together in a stable way and single electrons do not.
Could the pairing of electrons somehow account for the stable orbits? No, that isn't enough. Maxwell's Equations prove that no electron can ever have a stable orbit.
It was observed that sometimes electrons in atoms do radiate. When they change from one stable situation to another, they behave just like Maxwell's Equations would predict, they radiate exactly as they would if they were revolving around an atom and radiating, and falling. They radiate just enough to get from one stable configuration to another, and then they stop.
Nobody could solve the problem. There was no possible way for atoms to exist, and yet they do. And there was no way to get more information about how it happens. They just sit there. Anything you do to find out what they're doing, changes what they're doing.
Then they came up with Quantum Mechanics. It statistically describes what is happening. It makes no attempt to explain how it is that Maxwell's Equations are always perfectly correct but yet atoms exist. It simply describes what happens, and predicts what will happen averaged out over a whole lot of examples.
Quantum mechanics does not trace out any orbits. It provides a description of a lot of places an electron could be. They are compatible with an electron tracing out one of a lot of orbits, but they don't say the electron is actually orbiting. It just happens to be in those places. Maybe it isn't orbiting at all. Who knows? Maybe electrons have some way to orbit without falling, or maybe they don't move. QM works either way.
Traditionally magnets could be explained by electrons in orbits. In magnets some of the electrons all get lined up at the same angle so their magnetic force doesn't cancel. But if electrons can't orbit that doesn't work. Maybe each electron is always rotating, and it's the rotations adding up that make a magnet work.
But what about Maxwell's Equations that say electrons have to orbit? I dunno. That's classical physics. QM works, and that's good enough. Classical physics had something wrong with it, and nobody ever figured out how to make it work right.
So do free electrons have a minimal velocity? No. There's no reason to think so.
At low temperatures would electrons move around atoms at the same speed? It can be argued that they don't move around atoms at all. It's a classical concept anyway. We don't have to think about electrons moving around atoms.
The ground state kinetic energy depends on the external potential. It has nothing to do with intrinsic velocity or temperature.
If you solve the Schrödinger equation for an electron in an atom there is no temperature dependence. The ground state has 13.6 eV kinetic energy. For the free particle solution the lowest energy state corresponds to an infinite wavelength with zero kinetic energy (the velocity is certain but position maximally indeterminate).
