unambiguous context-free languages and complementation
Both questions turn out to have negative answers, as shown in [this][1] article.
In particular, the authors construct


An unambiguous context-free language whose complement is not context-free.
An inherently ambiguous (i.e., non-unambiguous) context-free language whose complement is context-free.


Consequently, there is no relationship between the notion of unambiguity for context-free languages and the complementation operation.

[1] The Independence of Inherent Ambiguity From Complementedness Among Context-Free Languages" by Hibbard and Ullian, JACM, Volume 13 Issue 4, Oct. 1966, Pages 588-593
 https://dl.acm.org/citation.cfm?doid=321356.321366