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We present measurements of secondary cosmic microwave background (CMB) anisotropies and cosmic infrared background (CIB) fluctuations using data from the South Pole Telescope (SPT) covering the complete 2540 deg2 SPT-SZ survey area. Data in the three SPT-SZ frequency bands centered at 95, 150, and 220 GHz, are used to produce six angular power spectra (three single-frequency auto-spectra and three cross-spectra) covering the multipole range 2000 < ℓ < 11, 000 (angular scales 5′ ≳ θ ≳ 1′). These are the most precise measurements of the angular power spectra at ℓ > 2500 at these frequencies. The main contributors to the power spectra at these angular scales and frequencies are the primary CMB, CIB, thermal and kinematic Sunyaev-Zel'dovich effects (tSZ and kSZ), and radio galaxies. We include a constraint on the tSZ power from a measurement of the tSZ bispectrum from 800 deg2 of the SPT-SZ survey. We measure the tSZ power at 143 GHz to be and the kSZ power to be . The data prefer positive kSZ power at 98.1% CL. We measure a correlation coefficient of between sources of tSZ and CIB power, with ξ < 0 disfavored at a confidence level of 99.0%. The constraint on kSZ power can be interpreted as an upper limit on the duration of reionization. When the post-reionization homogeneous kSZ signal is accounted for, we find an upper limit on the duration Δz < 5.4 at 95% CL. [ABSTRACT FROM AUTHOR]