The row of the character table of $S_n$ corresponding to the trivial representation has all entries positive, and by orthogonality clearly it is the only one like this.

Is it true that for $n\gg 0$, the row of the character table of $S_n$ corresponding to the sign representation has the most negative entries of any row (about half of them)?

Note that for $S_4$, in addition to the sign representation row there are two other rows which also have 2 negative entries.

Possibly the fact that row sums are positive is relevant here.