[Abridged] We calculate the structural evolution and nucleosynthesis of a grid of models covering the metallicity range: -6.5 < [Fe/H] < -3.0 (plus Z=0), and mass range: 0.85 < M < 3.0 Msun, amounting to 20 stars in total. In this paper, the first of a series describing and analysing this large data set, we present the resulting stellar yields. Many of the models experience violent nuclear burning episodes not seen at higher metallicities. We refer to these events as `Dual Flashes. These events have also been reported by previous studies. Some of the material processed by the Dual Flashes is dredged up causing significant surface pollution with a distinct chemical composition. We also analyse the yields in terms of C and N, comparing them to the observed CEMP abundances. At the lowest metallicities ([Fe/H] < -4.0) we find the yields to contain ~1 to 2 dex too much carbon, in agreement with all previous studies. At higher metallicities ([Fe/H] = -3.0), where the observed data set is much larger, all our models produce yields with [C/Fe] values consistent with those observed in the most C-rich CEMPs. However it is only the low-mass models that undergo the Dual Shell Flash (which occurs at the start of the TPAGB) that can best reproduce the C and N observations. Normal Third Dredge-Up can not reproduce the observations because at these metallicities intermediate mass models (M > 2 Msun) suffer HBB which converts the C to N thus lowering [C/N] well below the observations, whilst if TDU were to occur in the low-mass (M < 1 Msun) models (we do not find it to occur in our models), the yields would be expected to be C-rich only, which is at odds with the `dual pollution of C and N generally observed in the CEMPs.
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