We carried out extremely sensitive Submillimeter Array (SMA) 340 GHz (860 micron) continuum imaging of a complete sample of SCUBA 850 micron sources (>4 sigma) with fluxes >3 mJy in the GOODS-N. Using these data and new SCUBA-2 data, we do not detect 4 of the 16 SCUBA sources, and we rule out the original SCUBA fluxes at the 4 sigma level. Three more resolve into multiple fainter SMA galaxies, suggesting that our understanding of the most luminous high-redshift dusty galaxies may not be as reliable as we thought. 10 of the 16 independent SMA sources have spectroscopic redshifts (optical/infrared or CO) to z=5.18. Using a new, ultradeep 20 cm image obtained with the Karl G. Jansky Very Large Array (rms of 2.5 microJy), we find that all 16 of the SMA sources are detected at >5 sigma. Using Herschel far-infrared (FIR) data, we show that the five isolated SMA sources with Herschel detections are well described by an Arp 220 spectral energy distribution template in the FIR. They also closely obey the local FIR-radio correlation, a result that does not suffer from a radio bias. We compute the contribution from the 16 SMA sources to the universal star formation rate (SFR) per comoving volume. With individual SFRs in the range 700-5000 solar masses per year, they contribute ~30% of the extinction-corrected ultraviolet-selected SFR density from z=1 to at least z=5. Star formation histories determined from extinction-corrected ultraviolet populations and from submillimeter galaxy populations only partially overlap, due to the extreme ultraviolet faintness of some submillimeter galaxies. ; Comment: 26 pages, minor changes to match published version