1) That depends also on what tensor fitting option you've chosen. The most recent fitting options are iWLLS (iterative weighted)
and REKINDLE (outlier rejection)
. These work very similarly for DTI and DKI, with the only difference being the signal equation used in the fitting: DTI uses the second order tensor with 6 independent components (PJ Basser et al., 1994); DKI expands this by also fitting a fourth order tensor with 15 independent components (JH Jensen, MRM 2005). For both DTI and DKI, the first eigenvector from the second order tensor is used for tractography, as this describes the principal diffusion direction.
2) For both DTI and DKI the second order tensor is fit, and the first eigenvector is used. Also fitting the kurtosis tensor does affect the diffusion tensor, see Tax et al., ISMRM 2012, p3629. This is not working well with crossing fibres. Although DKI-derived dODFs have been proposed to resolve crossing fibres, the performance of these w.r.t. other crossing fibre models such as CSD haven't been evaluated thoroughly (to my knowledge). Anyway, ExploreDTI does not calculate the DKI-ODFs.
3) I would suggest not reordering the data at all. The averaging of b=0-images does not add anything and might only cause uncertainty if one or more are corrupted in a minor or major way.
Although I realise the following may be difficult if your datasets are already acquired, but I would strongly suggest to scan the multiple b=0-images not all at the beginning but spread out through your acquisition. My most recent paper that's just accepted in MRM demonstrates the presence of signal drift in dMRI data as a decrease in signal intensity as the acquisition progresses. This can be corrected after acquisition when multiple b=0-images are acquired spaced throughout the acquisition. It's a continuation of the ISMRM abstract I presented at the 2014 meeting (Vos et al., ISMRM 2014, p4460), and I'll post the link to the paper here as soon as it's online.