If only finite many cuts are finite, then only finitely many cuts are co-finite, which is a contradiction to the fact there are infinitely many cuts and each is a subset of $A$. The same argument shows that there cannot be only finitely many infinite cuts.
In Cohen's first model showing that ZF is consistent with the negation of AC was given by adding an iDf set of reals, which can be linearly ordered (but not well ordered). In particular in this model every set can be linearly ordered (for a slightly more detailed survey, see my answer here).
A note on consistency strength:
From the assumption that ZF is consistent you have that ZFC is consistent, and by forcing you have that "ZF+There is an iDf set+Every set if linearly orderable" is consistent, and by a different forcing you have that "ZF+Amorphous" is consistent. The last two clearly proving the consistency of ZF (if indeed they are consistent).
However, if we measure the consistency by how much we contradict the axiom of choice... well, in this case just as "axiom of countable choice" is stronger than "Every infinite set is Dedekind infinite" (i.e. the former assertion proves the latter over ZF), we have that:
"There exists an amorphous set" proves "There exists an iDf set", while the opposite is not true, as witnessed by the consistency of "ZF+There exists iDf set+Every set can be linearly ordered", since the last assertion is inconsistent with amorphous sets but still consistent with iDf sets.
And by its definition an iDf set which is not amorphous can be written as the disjoint union of two infinite sets. Therefore assuming that there exists an iDf set, but there are no amorphous sets is enough to ensure that every infinite set splits into two infinite sets. Whether or not this implies any other forms of choice (multiple choice, finite choice, choice from pairs, choice from well orderable sets, choice from a well orderable collection of well ordered sets, etc etc.), I do not know the answer for that. I'd be happy to look into this question, but this may take a few days due to prior engagements I have.
Lastly, two papers which may be interesting to this topic of conversation:
J.K.Truss, Classes of Dedekind Finite Cardinals, Fundamenta Mathematicae 84, 187-208, 1974.
In which seven notions of Dedekind finite cardinals (five in addition to finite sets and the one mentioned in my answer, which is the "canonical type" of infinite Dedekind finite cardinals). The paper is not very technical (I think) and I think that one can understand most of the results given without deep background in forcing or permutation models.
J.K.Truss, The structure of amorphous sets, Annals of Pure and Applied Logic, 73 (1995), 191-233.
This paper deals with amorphous sets, it is longer and more difficult than the previous one.
(I had to dig quite deep into the internet to find these online, and I cannot recall where I had found both papers. Many, many many thanks to Theo for finding the links.)