Once again, let me try to clarify what I think you're not getting. Quantum mechanics is exactly that, mechanics of particles. Just like Newtonian mechanics, but for quantum particles instead of macroscopic. The same way in classical physics you use Newton's laws to describe the motion of charged (or any, really) particles, in modern physics you use quantum mechanics. It doesn't "replace" electromagnetism, they're separate things. A bad metaphor, but it's a little like saying cardiology replaces anatomy.

Yes, electromagnetism is fundamental. No, you can't derive it from quantum mechanics — you'd use quantum field theory for that (specifically quantum electrodynamics).

Coulomb's law and Maxwell's equations are classical physics, but not mechanics. They don't describe the motion of particles, they describe forces and fields. And you use them all the time in quantum mechanics. Pick up any intro to quantum physics textbook, and you'll see them everywhere.

All that to say again that this meme is comparing Newton's law of gravitation with Coulomb's law... So why keep insisting on bringing quantum mechanics into this? It really doesn't figure into the meme at all.