Barriers to separating other complexity classes Do Natural Proofs, Relativization and Algebrization also affect separation of other complexity classes like $L\neq NL\neq NP\neq coNP \neq PH\neq PSPACE$ etc?

For instance natural proofs barrier should affect any proof of $NP\neq CoNP$ since it will separate $P\neq NP$. However relation between $NP$ and $CoNP$ does not seem to have much with OWFs as compared to relation between $P$ and $NP$. So do natural proofs affect the stronger separation of $NP\neq CoNP$?
There are (at least) two areas where existing barriers have little to say:

ACC Lower Bounds There is no known barrier to proving that TC0 is not in (non-uniform) ACC -- other than the possibility that the separation may be false. It's unclear whether the Natural Proofs barrier should apply to ACC. The question boils down to: should we expect there to be pseudorandom functions implementable in ACC?

LOGSPACE vs NP As pointed out by Fortnow, the existing oracle mechanisms for space-bounded computation do not seem to present a real barrier to LOGSPACE vs NP. To my knowledge, the known oracle models which yield a collapse of LOGSPACE and NP also collapse ALTERNATING LOGSPACE (i.e., P) and ALTERNATING POLYTIME (i.e., PSPACE), hence these oracles treat alternating computational models inconsistently with reality (since LOGSPACE is not equal to PSPACE).