  We study the small population of z>2.7 quasars detected by GALEX, whose
far-UV emission is not extinguished by intervening HI Lyman limit systems.
These quasars are of particular importance to detect intergalactic HeII
absorption along their sightlines. We correlate verified z>2.7 quasars to the
GALEX GR4 source catalog, yielding 304 S/N>3 sources. However, ~50% of these
are only detected in the GALEX NUV band, signaling the truncation of the FUV
flux by low-redshift Lyman limit systems. We exploit the GALEX UV color to cull
the most promising targets for follow-up studies, with blue (red) GALEX colors
indicating transparent (opaque) sightlines. Monte Carlo simulations indicate a
HeII detection rate of ~60% for quasars with FUV-NUV<1 at z<3.5, a ~50%
increase over GALEX searches that do not include color information. We regard
52 quasars detected at S/N>3 to be most promising for HST follow-up, with an
additional 114 quasars if we consider S/N>2 detections in the FUV. SDSS
provides just half of the NUV-bright quasars that should have been detected by
SDSS & GALEX. We revise the SDSS quasar selection function, finding that SDSS
systematically misses quasars with blue u-g<2 colors at 3<z<3.5 due to overlap
with the stellar locus in color space. Our color-dependent SDSS selection
function naturally explains the inhomogeneous u-g color distribution of SDSS
quasars with redshift and the color difference between color-selected and
radio-selected SDSS quasars. Moreover, it yields excellent agreement between
the observed and the predicted number of UV-bright SDSS quasars. We confirm our
previous claims that SDSS preferentially selects 3<z<3.5 quasars with
intervening HI Lyman limit systems. Our results imply that broadband optical
color surveys for 3<z<3.5 quasars have likely underestimated their space
density by selecting IGM sightlines with an excess of strong HI absorbers.
