As tests of QCD in the bottomonium system, we give the most accurate results to date for the ground-state hyperfine splitting and the $\Upsilon$ leptonic width from full lattice QCD. These quantities are both accurately known from experiment, so can provide a good test of $b$ physics, but previous lattice results have been rather imprecise. We also test the impact on these quantities of the $b$ quark’s electric charge. Our results are: $M_{\Upsilon} -M_{\eta_b} = 57.5(2.3)(1.0) \,\mathrm{MeV}$ (where
the second uncertainty comes from neglect of quark-line disconnected correlation
functions) and decay constants, $f_{\eta_b}=724(12)$ MeV and
$f_{\Upsilon} =677.2(9.7)$ MeV, giving
$\Gamma(\Upsilon \rightarrow e^+e^-) = 1.292(37)(3) \,\mathrm{keV}$.
We also give a new determination of the ratio of the masses for $b$ and $c$ quarks that is completely nonperturbative in lattice QCD and includes the calculation of QED effects for the first time. This gives a result for the $b$ quark mass of $\overline{m}_b(\overline{m}_b,n_f=5) = 4.202(21) $ GeV.