  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) 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) 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.
