Usually in set theory, Borel sets are much more nicely behaved than arbitrary sets of reals. One reason for this is Borel determinacy, which immediately yields measurability, Baireness, and the perfect set property for Borel sets; another reason (not totally distinct from the first) is that Borel sets enjoy a number of nice absoluteness properties. General nicenesses like these lead to, for example, the statement that the theory of Borel equivalence relations is largely immune to set-theoretic independence results (Parts of Set Theory immune to independence).

However, in 2008 Arnold Miller (http://arxiv.org/abs/0806.1957) showed that it is consistent with ZF that there is a Borel set which is infinite but Dedekind-finite (that is, admits no non-surjective self-injection). In fact, Miller's example is $F_{\sigma\delta}$, and he shows this is minimal: every Dedekind-finite $G_{\delta\sigma}$ set is finite. Miller notes that it is unknown whether there is an infinite Borel set with no surjection onto it union one additional element, or whether there is an infinite Borel set whose subsets are all countable or co-countable (called "quasi-amorphous").

My question is:

What are some other set-theoretically "bad" behaviors Borel sets can consistently have in $ZF$?

(To be somewhat precise, here "bad" behavior of Borel sets is behavior ruled out by $ZFC+$ some large cardinal axioms. When I say "consistently," I ideally mean "consistently relative to $ZFC$," but I would also be happy with examples which require some extra consistency strength.)

In particular, I suspect that most such results arise from violations of choice. So I would be particularly interested in:

What are some pathological properties Borel sets can have, consistently in $ZFC$?

This latter question of course takes on the pro-large cardinal perspective very forcefully, since in order to argue that a given behavior consistent with $ZFC$ is pathological we need to step outside $ZFC$ for our perspective.