So, there are various theorems that show that you can represent a distributive lattice as some sort of lattice of sets (birkhoff, stone, priestley etc).
Are there any theorems that provide representations of any kind for any species of non-distributive lattice? (for example, representation of orthomodular lattices)
There is Alasdair Urquart's topological representation of bounded lattices, in general, and it includes a not-nice-at-all representation of surjective homomorphisms.
This representation specializes to Priestley's one in case the lattice is distributive.
You can find it on the paper
A topological representation theory for lattices
published on Algebra Universalis, 1978, pages 45-58.
If you don't have access to that, I can sketch that representation (without proofs, of course).
Update. Actually, I already sketched that representation some time ago, in this answer.