We present cosmological constraints from the analysis of two-point correlation functions between galaxy positions and galaxy lensing measured in Dark Energy Survey (DES) Year 3 data and measurements of cosmic microwave background (CMB) lensing from the South Pole Telescope (SPT) and Planck. When jointly analyzing the DES-only two-point functions and the DES cross-correlations with SPT+Planck CMB lensing, we find ωm=0.344±0.030 and S8σ8(ωm/0.3)0.5=0.773±0.016, assuming ΛCDM. When additionally combining with measurements of the CMB lensing autospectrum, we find ωm=0.306-0.021+0.018 and S8=0.792±0.012. The high signal-to-noise of the CMB lensing cross-correlations enables several powerful consistency tests of these results, including comparisons with constraints derived from cross-correlations only, and comparisons designed to test the robustness of the galaxy lensing and clustering measurements from DES. Applying these tests to our measurements, we find no evidence of significant biases in the baseline cosmological constraints from the DES-only analyses or from the joint analyses with CMB lensing cross-correlations. However, the CMB lensing cross-correlations suggest possible problems with the correlation function measurements using alternative lens galaxy samples, in particular the redmagic galaxies and high-redshift maglim galaxies, consistent with the findings of previous studies. We use the CMB lensing cross-correlations to identify directions for further investigating these problems.