How does interval between us (e.g earth) and the spacetime outside of observable universe (above the event horizon) can be described in terms of timelike and spacelike intervals?
Is it just spacelike separation?
(I'm curious because it seems that space outside EH moves with speed >c from my reference frame and it's unusual)
In general relativity, we can't in general define a spacetime interval between two given events. You would also have to specify the path along which the metric was to be integrated. You can say you want a geodesic path, but in general that doesn't uniquely determine the path.
Normally we would describe this relationship in terms of light cones. We would say that a certain point in spacetime is outside of our past light cone.
(I'm curious because it seems that space outside EH moves with speed >c from my reference frame and it's unusual)
This is not really a correct description. Space doesn't move, and general relativity doesn't have global frames of reference, only local ones: How do frames of reference work in general relativity, and are they described by coordinate systems? GR doesn't generall give a unique, preferred definition of the velocity of object A relative to object B. In the context of a cosmological model, you can talk about the rate at which the proper distance between two comoving world-lines increases. By this definition, there are galaxies we observe that are now and always have been receding from us at a velocity greater than c. See At what speed does our universe expand?
