The effects of Ni doping in Eu(Co1-xNix)(2-y)As-2 single crystals with x = 0 to 1 grown out of self-flux are investigated via crystallographic, electronic transport, magnetic, and thermal measurements. All compositions adopt the body-centered-tetragonal ThCr2Si2 structure with space group I4/mmm. We also find 3%-4% of randomly distributed vacancies on the Co/Ni site. Anisotropic magnetic susceptibility chi(alpha) (alpha = ab, c) data versus temperature T show clear signatures of an antiferromagnetic (AFM) c-axis helix structure associated with the Eu+2 spins 7/2 for x = 0 and 1 as previously reported. The chi(alpha)(T) data for x = 0.03 and 0.10 suggest an anomalous 2q magnetic structure containing two helix axes along the c axis and in the ab plane, respectively, whereas for x = 0.75 and 0.82 a c-axis helix is inferred as previously found for x = 0 and 1. At intermediate compositions x = 0.2, 0.32, 0.42, 0.54, and 0.65, a magnetic structure with a large ferromagnetic (FM) c-axis component is found from magnetization versus field isotherms, suggested to be an incommensurate FM c-axis cone structure associated with the Eu spins, which consists of both AFM and FM components. In addition, the chi(T) and heat capacity C-p(T) data for x = 0.2-0.65 indicate the occurrence of itinerant FM order associated with the Co/Ni atoms with Curie temperatures from 60 to 25 K, respectively. Electrical resistivity rho(T) measurements indicate metallic character for all compositions with abrupt increases in slope on cooling below the Eu AFM transition temperatures. In addition to this panoply of magnetic transitions, Eu-151 Mossbauer measurements indicate that ordering of the Eu moments proceeds via an incommensurate sine amplitude-modulated structure with additional transition temperatures associated with this effect.