We present full spectral scans from 200-670$mu$m of 26 Class 0+I protostellar sources, obtained with $Herschel$-SPIRE, as part of the COPS-SPIRE Open Time program, complementary to the DIGIT and WISH Key programs. Based on our nearly continuous, line-free spectra from 200-670 $mu$m, the calculated bolometric luminosities ($L_{rm bol}$) increase by 50% on average, and the bolometric temperatures ($T_{rm bol}$) decrease by 10% on average, in comparison with the measurements without Herschel. Fifteen protostars have the same Class using $T_{rm bol}$ and $L_{rm bol}$/$L_{rm submm}$. We identify rotational transitions of CO lines from J=4-3 to J=13-12, along with emission lines of $^{13}$CO, HCO$^+$, H$_{2}$O, and [CI]. The ratios of $^{12}$CO to $^{13}$CO indicate that $^{12}$CO emission remains optically thick for $J_{rm up}$ < 13. We fit up to four components of temperature from the rotational diagram with flexible break points to separate the components. The distribution of rotational temperatures shows a primary population around 100 K with a secondary population at $sim$350 K. We quantify the correlations of each line pair found in our dataset, and find the strength of correlation of CO lines decreases as the difference between $J$-level between two CO lines increases. The multiple origins of CO emission previously revealed by velocity-resolved profiles are consistent with this smooth distribution if each physical component contributes to a wide range of CO lines with significant overlap in the CO ladder. We investigate the spatial extent of CO emission and find that the morphology is more centrally peaked and less bipolar at high-$J$ lines. We find the CO emission observed with SPIRE related to outflows, which consists two components, the entrained gas and shocked gas, as revealed by our rotational diagram analysis as well as the studies with velocity-resolved CO emission.
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