  The soft gamma repeater SGR 1806-20 is most famous for its giant flare from
2004, which yielded the highest gamma-ray flux ever observed on Earth. The
flare emphasized the importance of determining the distance to the SGR, thus
revealing the flare's energy output, with implications on SGRs energy budget
and giant flare rates. We analyze x-ray scattering echoes observed by Swift/XRT
following the 2006 August 6 intermediate burst of SGR 1806-20. Assuming
positions and opacities of the molecular clouds along the line-of-sight from
previous works, we derive direct constrains on the distance to SGR 1806-20,
setting a lower limit of 9.4 kpc and an upper limit of 18.6 kpc (90%
confidence), compared with a 6-15 kpc distance range by previous works. This
distance range matches an energy output of ~10^46 erg/s for the 2004 giant
flare. We further use, for the first time, the x-ray echoes in order to study
the dust properties in molecular clouds. Analyzing the temporal evolution of
the observed flux using a dust scattering model, which assumes a power-law size
distribution of the dust grains, we find a power-law index of
-3.3_{-0.7}^{+0.6} (1 sigma) and a lower limit of 0.1 micron (2 sigma) on the
dust maximal grain size, both conforming to measured dust properties in the
diffused interstellar medium (ISM). We advocate future burst follow-up
observations with Swift, Chandra and the planned NuSTAR telescopes, as means of
obtaining much superior results from such an analysis.
