The lambda calculus is an untyped language that is often extended with logical frameworks such as the vertices of the λ-cube. Is there something similar to it, but for interaction nets? What about interaction combinators?
The Structure of Interaction paper seems to be what you are asking for.
Like Martin said, nothing like an equivalent for the $\lambda$-cube has ever been developed for interaction nets or interaction combinators.
The only work that considers types for interaction nets is Lafont's original paper . (My own CONCUR 2005 paper also considers types but adds nothing new to what Lafont did). It is an extremely simple system: only basic types plus a polarity, no type constructors at all. Nevertheless, it is enough to obtain a desirable safety property: a well typed net cannot reduce to a net containing a "clash" (an active pair which does not have a corresponding reduction rule and is therefore "stuck").
Of course, since multiplicative linear logic proof nets are a particular system of interaction nets, it is defintely possible to equip certain systems of interaction nets with more complex types, using non-trivial type constructors. But nothing has ever been studied, mostly because the interest of such an endevour is unclear.
 Yves Lafont, Interaction Nets. In Proceedings of POPL, 1990.