Is there a theory of hypergraph minors? I could only find some attempts to define them at papers/theses, whose main topic was something else. What would be a useful definition? Does the hypergraph version of the Robertson–Seymour theorem hold?
I think both questions of the OP are answered in: Hypertree-depth and minors in hypergraphs (2012)
Let $H$ and $H′$ be hypergraphs. Then $H$ is a minor of $H′$ if $H$ can be obtained from $H′$ by a sequence of operations of the following kinds:
• vertex deletion,
• contraction of (the edge between) two vertices that are contained in a common hyperedge,
• addition of ahyperedge $e$ such that the set $e$ induces a clique in the underlying graph, and
• deletion of a proper subhyperedge.
For graphs, a famous result by Robertson and Seymour shows that testing for a fixed minor is solvable in cubic time. In contrast, testing for a fixed hypergraph minor can be NP-hard (section 6.4).