Could there be a connection between the way the earth revolves around the sun and the way an electron orbits a nucleus (besides that they attract each other with their masses)?

Answer

These are two different things though. The only similarity is that the planet’s orbit around the sun, and the electron’s behavior around the nucleus, are forces determined by fundamental forces.
For the planet this is only gravity, because the other fundamental forces are irrelevant : the electromagnetic force has no effect because the sun and the planets are both electrically neutrally charged, and the strong and weak nuclear forces only work on (sub) atomic scale.
With an electron the situation is reversed : there gravitation does not play a role, but it is the electromagnetic force that plays. The gravitational force between electrons is completely negligible. Gravitation and EM force also work differently. Gravitation acts between all masses and is always attractive, EM force only acts between charges (so it has no effect on neutral charges) and can be both attractive and repulsive.

The main difference between the two situations is the importance of quantum mechanical effects. which cannot be neglected in the case of the electron, unlike in the case of a planet around the sun. In astronomy you can really see the planet as a sphere of matter orbiting the sun, but in the case of an electron you have to see the electron partly as a matter particle, partly as a wave. A second consequence is that those electrons cannot just move in any “orbit”, while planets can in principle. The behavior of an electron is also strongly determined by the presence and behavior of the other electrons. This is also the case with planets, but relatively to a much lesser extent.

The representation of an atomic nucleus + electrons as a kind of mini-solar system is therefore too much of a simplification.