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In reply to your first remark, replacing projectors with density operators does not make much sense to me, but maybe there is a way to do it that is interesting. One might, for example, purify the density operator by using a reference system, still regarded as part of the closed system whose overall history one is trying to describe, assume trivial unitary time evolution for the reference system, as is sometimes done in quantum information discussions, and apply consistency by demanding that the off-diagonal elements of the Gell-Mann and Hartle decoherence functional vanish. This is vaguely similar to what I and two collaborators did in . We had a definite physical motivation for doing so, and I don’t know how it might work out in some other case.
Relative to your second remark. I remember looking at a paper which employed multiple times. Not closely enough to spot any technical difficulties, but I also didn’t see why it was of that much interest. If using a single time suffices for describing what one is interest it, what advantage is there to using several?
 Patrick J. Coles, Vlad Gheorghiu, Robert B. Griffiths,
“Collisional decoherence of tunneling molecules: a consistent histories treatment” Phys. Rev. A 86 (2012) 042111; arXiv:1205.6188
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