Nope, 2nd-order logic can discuss Nth order where N is an infinite ordinal, as well [for some class of describable ordinals]. Maybe the argument could be made that 2nd-order logic cannot discuss a universe containing all the ordinals, and in that case maybe we could argue that set theory can, and is therefore more powerful. But this is not clear.

Independently of that, we might also believe that category theory can describe things beyond set theory, because category theory regularly investigates categories which correspond to "large" sets (such as the set of all sets). There are ways to put category theory into set theory, but these translate the "large" sets into "small" sets such as Grothendeik universes, so we could still argue that the semantics of category theory is bigger.

However, even if this is the case, it might be that 2nd-order logic can encode category theory in the same way that it can encode 3rd-order logic. We can add axioms to 2nd-order logic which describe non-standard set theories containing "big" sets, such as NF. This may allow for an "honest" account of category theory, in which categories really can be defined on "big" sets such as the set of all sets.

I don't think that infinitary logic is the same as ω-order logic.

Third order logic talks about something that we don't understand, and we don't understand it in exactly the same way that the first-order aliens can't understand second order logic.

One of the things third order logic might be able to do is determine the number of steps a Turing will take before halting in a constant-time algorithm.