  We present Hubble Space Telescope (HST) ACS and WFC3 observations of SDSS
J1029+2623, a three-image quasar lens system produced by a foreground cluster
at z=0.584. Our strong lensing analysis reveals 6 additional multiply imaged
galaxies. We confirm the complex nature of the mass distribution of the lensing
cluster, with a bimodal distribution which deviates from the Chandra X-ray SB
distribution. The Einstein radius is estimated to be \theta_E=15.2" \pm 0.5"
for the quasar redshift of z=2.197. We derive a radial mass distribution from
the combination of strong lensing, HST/ACS weak lensing, and Subaru/Suprime-cam
weak lensing analysis results, finding a best-fit virial mass of
M_vir=(1.5+0.40-0.35) \times 10^14 h^-1 M_sun and a concentration parameter of
c_vir=25.7+14.1-7.5. The lensing mass estimate at the outer radius is smaller
than the X-ray mass estimate by a factor of ~2. We ascribe this large mass
discrepancy to shock heating of the intracluster gas during a merger, which is
also suggested by the complex mass and gas distributions and the high value of
the concentration parameter. In the HST image, we also identify a probable
galaxy, GX, in the vicinity of the faintest quasar image C. The inclusion of GX
explains the anomalous flux ratios between the quasar images. The morphology of
the highly elongated quasar host galaxy is also well reproduced. The best-fit
model suggests large total magnifications of 30 for the quasar and 35 for the
quasar host galaxy, and has an AB time delay consistent with the measured
value. (Abridged)
