  Joint analyses of cross-correlations between measurements of galaxy
positions, galaxy lensing, and lensing of the cosmic microwave background (CMB)
offer powerful constraints on the large-scale structure of the Universe. In a
forthcoming analysis, we will present cosmological constraints from the
analysis of such cross-correlations measured using Year 3 data from the Dark
Energy Survey (DES), and CMB data from the South Pole Telescope (SPT) and
Planck. Here we present two key ingredients of this analysis: (1) an improved
CMB lensing map in the SPT-SZ survey footprint, and (2) the analysis
methodology that will be used to extract cosmological information from the
cross-correlation measurements. Relative to previous lensing maps made from the
same CMB observations, we have implemented techniques to remove contamination
from the thermal Sunyaev Zel'dovich effect, enabling the extraction of
cosmological information from smaller angular scales of the cross-correlation
measurements than in previous analyses with DES Year 1 data. We describe our
model for the cross-correlations between these maps and DES data, and validate
our modeling choices to demonstrate the robustness of our analysis. We then
forecast the expected cosmological constraints from the galaxy survey-CMB
lensing auto and cross-correlations. We find that the galaxy-CMB lensing and
galaxy shear-CMB lensing correlations will on their own provide a constraint on
$S_8=\sigma_8 \sqrt{\Omega_{\rm m}/0.3}$ at the few percent level, providing a
powerful consistency check for the DES-only constraints. We explore scenarios
where external priors on shear calibration are removed, finding that the joint
analysis of CMB lensing cross-correlations can provide constraints on the shear
calibration amplitude at the 5 to 10% level.
