  Q0302-003 ($z=3.2860 \pm 0.0005$) was the first quasar discovered that showed
a He II Gunn-Peterson trough, a sign of incomplete helium reionization at $z >
2.9$. We present its HST/Cosmic Origins Spectrograph far-UV medium-resolution
spectrum, which resolves many spectral features for the first time, allowing
study of the quasar itself, the intergalactic medium, and quasar proximity
effects. Q0302-003 has a harder intrinsic extreme-UV spectral index than
previously claimed, as determined from both a direct fit to the spectrum
(yielding $\alpha_{\nu} = -0.8$) and the helium-to-hydrogen ion ratio in the
quasar's line-of-sight proximity zone. Intergalactic absorption along this
sightline shows that the helium Gunn-Peterson trough is largely black in the
range $2.87 < z < 3.20$, apart from ionization due to local sources, indicating
that helium reionization has not completed at these redshifts. However, we
tentatively report a detection of nonzero flux in the high-redshift trough when
looking at low-density regions, but zero flux in higher-density regions. This
constrains the He II fraction to be a few percent, suggesting helium
reionization has progressed substantially by $z \sim 3.1$. The Gunn-Peterson
trough recovers to a He II Ly$\alpha$ forest at $z < 2.87$. We confirm a
transmission feature due to the ionization zone around a $z = 3.05$ quasar just
off the sightline, and resolve the feature for the first time. We discover a
similar such feature possibly caused by a luminous $z = 3.23$ quasar further
from the sightline, which suggests that this quasar has been luminous for >34
Myr.
