Is there any hash function for which a preimage of a fixed element is always known?
Take any standard hash function family $\{h_s(\cdot)\}_s$, a point $c$ and a list of inputs $(x_s)_s$ and define $h'_s : x \mapsto h_s(x)$ if $x\neq x_s$, and $c$ otherwise. As you said, it does not contradict collision resistance or preimage resistance - in other words, you can prove that if $(h_s)_s$ is a family of (say) collision-resistant hash functions, then so is $(h'_s)_s$. If you have a collision on Merkle-Damgård applied to $(h'_s)_s$ without $IV$, then you are done. The existence of any collision-resistant hash function implies the existence of a hash with the properties you want, via this trivial method.