For example, a Hydrogen atom has the smallest nucleus and so the orbit of its electron should have a smaller circumference, but this does not conclude anything absolute about the speed of that electron compared to the speed of an electron orbiting in the 3rd or 4th shell of a much larger nucleus.

Electrons are not in classical orbits around atoms, they're in quantum mechanical states.

To continue with what Klaynos said, "speed of an electron" loses meaning in QM when discussing orbitals. You refer to the energy.

The closest I can think of is to employ the Bohr model, here we do have a notion of the velocity of an electron wizzing round the atomic nucleus. However, this model has lots of problems and I would consider the calculation of the said velocity to be very huristic, but it could be useful in some contexts.

 

The modern picture based on quantum mechanics does not give us a clear picture of an orbit of an electron (as a classical trajectory) nor the velocity of an electron in an orbital. There maybe ways of getting some semi-classical view of a velocity here, but again it would be very huristic and one would have to be very careful in interpreting it.

The (fuzzy) dimensions of an orbital decreases with more protons as these attract the electrons more strongly. The realtion is inversely proportional. This tells why the ionization energy of the K shell increases as the square of the number of protons.

 

The angular momentum of an orbital may be definite, at least around one direction. The speed is a less clear idea; by most definitions of speed, it's zero for all spherical shells.