This is also a comment, along the lines of usul's comment but a little different.  I think it is best not to phrase your question in terms of "no matter what input type is used" but "no matter how apathetic the voters are."  To be more clear, the way I think of Arrow's theorem is not that if you design a ballot with a total order then you are in trouble; rather, it's that <i>if voters have such sophisticated preferences that they are able to rank all the candidates in a total order</i> then you are in trouble.  What I think you're trying to get at with the partial orders is to ask, what if voters don't actually care about the candidates so much that they are able to rank them in strict order?  Intuitively, the more apathetic the voters are, the easier it should be to mollify them.

Thus the most general impossibility theorem would be something like, even if the voters don't care about any of the candidates at all, there's no way to select a satisfactory candidate, and now we can easily see that there trivially can't be a theorem that general.  If nobody cares, then just pick anybody, and there can't be any objection.

Somewhat less trivially, if every voter has just one candidate that they like and they are totally indifferent among the others, then picking the candidate with the most votes is going to be paradox-free according to almost any reasonable criteria for a satisfactory voting system.

So I think what you are asking for is, what is the most apathetic voter population that cannot always be appeased?  I'm a bit skeptical that there can be a unique "optimal" impossibility theorem in this sense, but maybe there is.