Let $X$ be an abstract not necessarily compact orientable CR manifold of dimension $2n-1$, $n\geqslant2$. Let
$\Box^{(q)}_{b}$ be the Gaffney extension of Kohn Laplacian for $(0,q)$--forms. We show that the spectral
function of $\Box^{(q)}_b$ admits a full asymptotic expansion on the non-degenerate part of the Levi form.
As a corollary, we deduce that if $X$ is compact and the Levi form is non-degenerate of constant signature on $X$,
then the spectrum of $\Box^{(q)}_b$ in $]0,\infty[$ consists of point eigenvalues of finite multiplicity.
Moreover, we show that a certain microlocal conjugation of the associated Szeg\H{o} kernel admits an asymptotic expansion
under a local closed range condition. As applications, we establish the Szeg\H{o} kernel asymptotic expansions
on some weakly pseudoconvex CR manifolds and on CR manifolds with transversal CR $S^1$ actions.
By using these asymptotics, we establish some local embedding theorems on CR manifolds and we give
an analytic proof of a theorem of Lempert asserting that a compact strictly pseudoconvex
CR manifold of dimension three with a transversal CR $S^1$ action can be CR embedded
into $\mathbb C^N$, for some $N\in\mathbb N$.