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Charge density is the time-like component of the current 4-vector and charge is a Lorentz scalar. Charge is defined as the integral of the charge density over a space-like hypersurface. The proof that charge is a Lorentz scalar is actually quite simple, given the 4-divergence of the current 4-vector. It's basically just an application of Stokes' theorem.

Yes, you're right. I can't find my copy of Jackson, so I'm not sure what the alternative theory is that reduces to Coulomb's law in the nonrelativistic limit.