We explore extensions to the $Lambda$CDM cosmology using measurements of the cosmic microwave background (CMB) from the recent SPT-SZ survey, along with data from WMAP7 and measurements of $H_0$ and BAO. We check for consistency within $Lambda$CDM between these datasets, and find some tension. The CMB alone gives weak support to physics beyond $Lambda$CDM, due to a slight trend relative to $Lambda$CDM of decreasing power towards smaller angular scales. While it may be due to statistical fluctuation, this trend could also be explained by several extensions. We consider running index (nrun), as well as two extensions that modify the damping tail power (the primordial helium abundance $Y_p$ and the effective number of neutrino species $N_{rm eff}$) and one that modifies the large-scale power due to the ISW effect (the sum of neutrino masses $sum m_ u$). These extensions have similar observational consequences and are partially degenerate when considered simultaneously. Of the 6 one-parameter extensions considered, we find CMB to have the largest preference for nrun with -0.046<nrun<-0.003 at 95% confidence, which strengthens to a 2.7$sigma$ indication of nrun<0 from CMB+BAO+$H_0$. Detectable non-zero nrun is difficult to explain in the context of single-field, slow-roll inflation models. We find $N_{rm eff}=3.62pm0.48$ for the CMB, which tightens to $N_{rm eff}=3.71pm0.35$ from CMB+BAO+$H_0$. Larger values of $N_{rm eff}$ relieve the mild tension between CMB, BAO and $H_0$. When the SZ selected galaxy cluster abundances ($rm{SPT_{CL}}$) data are also included, we obtain $N_{rm eff}=3.29pm0.31$. Allowing for $sum m_ u$ gives a 3$sigma$ detection of $sum m_ u$>0 from CMB+BAO+$H_0$+$rm{SPT_{CL}}$. The median value is $(0.32pm0.11)$ eV, a factor of six above the lower bound set by neutrino oscillation observations. ... [abridged]
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