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Using the EVEREST photometry pipeline, we have identified 74 candidate ultra-short-period planets (orbital period P<1 d) in the first half of the K2 data (Campaigns 0-8 and 10). Of these, 33 candidates have not previously been reported. A systematic search for additional transiting planets found 13 new multi-planet systems, doubling the number known and representing a third (32%) of USPs. We also identified 30 companions, which have periods from 1.4 to 31 days (median 5.5 d). A third (36 of 104) of the candidate USPs and companions have been statistically validated or confirmed, 10 for the first time, including 7 USPs. Almost all candidates, and all validated planets, are small (radii Rp<=3 R_E) with a median radius of R_p=1.1 R_E; the validated and confirmed candidates have radii between 0.4 R_E and 2.4 R_E and periods from P=0.18 to 0.96 d. The lack of candidate (a) ultra-hot-Jupiters (R_p>10 R_E) and (b) short-period desert (3<=Rp<=10 R_E) planets suggests that both populations are rare, although our survey may have missed some of the very deepest transits. These results also provide strong evidence that we have not reached a lower limit on the distribution of planetary radius values for planets at close proximity to a star, and suggest that additional improvements in photometry techniques would yield yet more ultra-short-period planets. The large fraction of USPs in known multi-planet systems supports origins models that involve dynamical interactions with exterior planets coupled to tidal decay of the USP orbits.