  We present STIS 7km/s resolution data of the Ly-a forest toward PKS0405-123
(z=0.574). We make two samples. The strong sample (60 systems) has column
density logNHI>13.3 over 0.002<z<0.423, and the weak sample (44) has
logNHI>13.1 at 0.020<z<0.234; 7 absorbers show metals, all with OVI, but often
offset in velocity from HI. The strong sample Doppler parameters show
<b>=47+-22km/s; for the weak, <b>=44+-21km/s. Line blending and s/n effects
likely inflate b. The redshift density dN/dz is consistent with previous, lower
resolution results for logNHI>14.0. For 13.1<logNHI<14.0, we find a dN/dz
overdensity of ~0.2-0.3 dex at 0.127<z<0.234, which we believe arises from
cosmic variance. We find Ly-a clustering over Delta v<250km/s for logNHI>13.3,
consistent with a numerical model. There is a void in the strong sample at
0.0320<z<0.0814. We detect radial velocity correlations for Delta v<250km/s
between Ly-a absorbers and 39 galaxies in a 10' field. The correlation grows
with minimum NHI, peaking for logNHI ~> 13.5-14.0. The strength is similar to
that of the galaxy-galaxy correlation for our sample, implying that such Ly-a
systems have mass log M/Msun= 11.3+1.0-0.6. Including lower column density
systems in the sample shows correlations only out to Delta v <125km/s, as would
be expected for smaller density perturbations. Galaxy counts and local HI
column density appear correlated, peaking at ~5000km/s. We also give column
densities for Galactic species. (Abridged)
