Is there a "lambda cube" for interaction nets?
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 [1].  (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.

[1] Yves Lafont, Interaction Nets.  In Proceedings of POPL, 1990.