sorry pointwise*

No, look at the complete lattice of the powerset of natural numbers, take the sequence x(n) = {1,...,n}. Then liminf x = limsup x = N but (E,N] where E={0,2,4,...} is an neighborhood of N containing none of the x(n)

Question: if you take a complete lattice, generate the topology via open intervals in the lattice, do topological limit and the order limit always coincide?

Oh nice yea! Liminf and limsup are well defined in any complete lattice. And if they are equal, you can define that to be the limit. No topology needed!

And I guess saying it's bounded say by [-b,b] then that's implicitly using that [-b,b] is a complete lattice

So it holds for extended real valued random variables but not real valued, unless it's assumed the set {X_i(omega)|i in I} is bounded for all omega