  We study the $1$-level density and the pair correlation of zeros of quadratic
Dirichlet $L$-functions in function fields, as we average over the ensemble
$\mathcal{H}_{2g+1}$ of monic, square-free polynomials with coefficients in
$\mathbb{F}_q[x]$. In the case of the $1$-level density, when the Fourier
transform of the test function is supported in the restricted interval
$(\frac{1}{3},1)$, we compute a secondary term of size $q^{-\frac{4g}{3}}/g$,
which is not predicted by the Ratios Conjecture. Moreover, when the support is
even more restricted, we obtain several lower order terms. For example, if the
Fourier transform is supported in $(\frac{1}{3}, \frac{1}{2})$, we identify
another lower order term of size $q^{-\frac{8g}{5}}/g$. We also compute the
pair correlation, and as for the $1$-level density, we detect lower order terms
under certain restrictions; for example, we see a term of size $q^{-g}/g^2$
when the Fourier transform is supported in $(\frac{1}{4},\frac{1}{2})$. The
$1$-level density and the pair correlation allow us to obtain non-vanishing
results for $L(\frac12,\chi_D)$, as well as lower bounds for the proportion of
simple zeros of this family of $L$-functions.
