Are there corollaries (or generalised versions) of Arrow's theorem covering obvious fixes? For a given set of alternatives $X=\{x_1,\dots,x_n \}$, let each individual have a total and transitive preference order among the alternatives $X$. The goal would be to have a system or rule to convert these individual preferences into community-wide preference order.
A simple randomised rule would be to choose an individual randomly, and declare his preference order to be the community-wide preference order. Such a rule could still be unfair, if certain individuals would be chosen with a higher probability than others.
For deterministic rules, Arrow's theorem shows that under reasonable conditions for the rule, the only possibility is to take the preference order of a single individual as the community-wide preference order. But for randomised rules, there might be other possible rules which also satisfy all other conditions. A randomised quick sort might be such a rule, where an alternative and an individual are selected randomly, the preference order of the individual then partitions the alternatives into three sets, and two of those sets are further sorted by recursively applying the same rule.
Another obvious modification would be that in addition to a preference order, each individual also partitions the set of alternatives into three sets, the ones he would really like, the ones he would really dislike, and the ones about which he is indifferent. Of course this partition would have to be compatible with the preference order of the individual.

Are there corollaries (or generalised versions) of Arrow's theorem which would describe the possible rules that satisfy reasonable conditions similar to those from from Arrow's theorem, if such obvious "fixes" as described above are present (i.e. allowing randomised rules and allowing to explicitly state which alternatives are desired and which are undesired).

 A: The short answer is there is a whole literature on the "fixes" you propose, and others. It's hard to point you in one particular direction, other than inviting you to have a look at the two volumes of the Handbook of Social Choice and Welfare (https://www.elsevier.com/books/handbook-of-social-choice-and-welfare/arrow/978-0-444-82914-6 and https://www.elsevier.com/books/handbook-of-social-choice-and-welfare/arrow/978-0-444-50894-2).
For example, your second "fix" is related to a domain restriction called "dichotomous" preferences (or maybe "trichotomous" in the case you describe). What more can and cannot be done under this kind of restrictions is discussed (among many other places) in:

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*Gaertner, Wulf. "Domain restrictions." Handbook of social choice and welfare 1 (2002): 131-170., (https://www.sciencedirect.com/science/article/pii/S1574011002800078)


*Maniquet, François, and Philippe Mongin. "Approval voting and Arrow’s impossibility theorem." Social Choice and Welfare 44.3 (2015): 519-532. (https://link.springer.com/content/pdf/10.1007/s00355-014-0847-2.pdf)
See also this reference for an example of a discussion of other domain restrictions,

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*Brandl, Florian, and Felix Brandt. "Arrovian aggregation of convex preferences." Econometrica 88.2 (2020): 799-844. https://onlinelibrary.wiley.com/doi/full/10.3982/ECTA15749?casa_token=5D5D--K_KeoAAAAA%3A1510ADZreY_PMY-Jif9NbDoxSeDUwlhWZooo0YvcE6tKCSDfkEOifT73WiA0x3PxpmwLZUgauVKHi571
