Just write down the definitions, and write down their negations.
A set $A$ in a metrix space $(X,d)$ is nowhere dense if the closure of $A$
has empty interior, that is, equivalently - its closure does not contain an open ball of the metric space.
A set $B$ is a metric space $(X,d)$ is everywhere dense if for every open set $O\subset X$, the intersection $B\cap O$ is not empty.
Consider now a set $A\subset X$. If the complement is not everywhere dense,
then there is some non-empty open set $O\subset X$ such that $O\cap A^c=\emptyset$. It follows that $O\subset A$, and so $A$ has non-empty interior, hence its closure has non-empty interior, hence it is not nowhere dense. It follows that if $A$ is nowhere dense, then $A^c$ is everywhere dense.
As for the converse, the question is whether the complement of a dense set is nowhere dense. This is not true: Take the rational points in $[0,1]$, they are dense. Their complement is the set of irrationals, whose closure is $[0,1]$, so it is not nowhere dense.
The second question was already answered in the comments.