Chickenshit and batshit seem to be extensions of chicken (scared) and batty/bats in the belfry, respectively.

I don't think I've come across dogshit personally so would read it as an extension of regular shit.

Horseshit is synonymous with bullshit I think, probably because in both cases there's a whole lot of it. Also by synonymy are euphemistic phrases like "manure" (and "guano" I think even though it's the wrong kind of shit).

Trying to think of other kinds of shit but catshit, turkeyshit, duckshit, sheepshit, pigshit all just sound like literal shit to me.