Is it known if $CFL \subseteq NSPACE(o(log^2(n)))$?
Yes. This is Theorem 4 in the following paper.

Philip M. Lewis II, Richard Edwin Stearns, Juris Hartmanis:
Memory bounds for recognition of context-free and context-sensitive languages. SWCT (FOCS) 1965: 191-202.

Edit: I am sorry. My answer is not helpful. I missed the "small o". Stupid... 
I am not sure what to do. Delete the answer?