Proof of homotopy invariance: This follows from a base change/Kunneth type statement and the calculation of the cohomology of $\mathbb A^1$.

Specifically, Lemma 7.6.7 of Lei Fu's etale cohomology theory, specialized to $S$ a point, $f$ the map from $X$ to a point, $g$ the map from $\mathbb A^1$ to a point, so that $g' = p$, $K$ an arbitrary complex on $X$, and $L$ the constant sheaf, implies that $p_* p^* K = K \otimes f^* g_* \mathbb Q_\ell$ as soon as $K$ is strongly locally acyclic relative to $f$, which it is by Lemma 9.3.4.

Then $g_* \mathbb Q_\ell$ is the cohomology of the affine line, which is simply $\mathbb Q_\ell$, so $K \otimes f^* g_* \mathbb Q_\ell = K \otimes f^* \mathbb Q_\ell = K \otimes \mathbb Q_\ell= K$ (or the same with a different set of coefficients.

Proof of stability: As Deligne notes in the very last paragraph, in the etale setting $p_\#$ is $p_!$ up to a shift and twist: For a smooth morphism, $p^*$ and $p^!$ agree up to a shift and twist, and $p^!$ has a left adjoint $p_!$, which thus agrees with $p_\#$ up to the dual shift and twist.

Since shifting and twisting are equivalences of categories, it suffices to check that $p_! s_*$ is an equivalence of categories. Now $s_*$ of any sheaf is compactly supported over the base $X$ (since a section is compact), which means $p_! s_* = p_* s_*$, and $p_* s_*$ is the identity, and thus an equivalence, by the Leray spectral sequence.

Not sure on the motivation.

Proof of homotopy invariance: This follows from a base change/Kunneth type statement and the calculation of the cohomology of $\mathbb A^1$.

Specifically, Lemma 7.6.7 of Lei Fu's etale cohomology theory, specialized to $S$ a point, $f$ the map from $X$ to a point, $g$ the map from $\mathbb A^1$ to a point, so that $g' = p$, $K$ an arbitrary complex on $X$, and $L$ the constant sheaf, implies that $p_* p^* K = K \otimes f^* g_* \mathbb Z_\ell$ as soon as $K$ is strongly locally acyclic relative to $f$, which it is by Lemma 9.3.4.

Then $g_* \mathbb Z_\ell$ is the cohomology of the affine line, which is simply $\mathbb Z_\ell$, so $K \otimes f^* g_* \mathbb Z_\ell = K \otimes f^* \mathbb Z_\ell = K \otimes \mathbb Z_\ell= K$.

Proof of stability: As Deligne notes in the very last paragraph, in the etale setting $p_\#$ is $p_!$ up to a shift and twist: For a smooth morphism, $p^*$ and $p^!$ agree up to a shift and twist, and $p^!$ has a left adjoint $p_!$, which thus agrees with $p_\#$ up to the dual shift and twist.

Since shifting and twisting are equivalences of categories, it suffices to check that $p_! s_*$ is an equivalence of categories. Now $s_*$ of any sheaf is compactly supported over the base $X$ (since a section is compact), which means $p_! s_* = p_* s_*$, and $p_* s_*$ is the identity, and thus an equivalence, by the Leray spectral sequence.

Not sure on the motivation.

I can give a proof of stability.

As Deligne notes in the very last paragraph, in the etale setting $p_\#$ is $p_!$ up to a shift and twist: For a smooth morphism, $p^*$ and $p^!$ agree up to a shift and twist, and $p^!$ has a left adjoint $p_!$, which thus agrees with $p_\#$ up to the dual shift and twist.

Since shifting and twisting are equivalences of categories, it suffices to check that $p_! s_*$ is an equivalence of categories. Now $s_*$ of any sheaf is compactly supported over the base $X$ (since a section is compact), which means $p_! s_* = p_* s_*$, and $p_* s_*$ is the identity, and thus an equivalence, by the Leray spectral sequence.

Proof of homotopy invariance: This follows from a base change/Kunneth type statement and the calculation of the cohomology of $\mathbb A^1$.

Specifically, Lemma 7.6.7 of Lei Fu's etale cohomology theory, specialized to $S$ a point, $f$ the map from $X$ to a point, $g$ the map from $\mathbb A^1$ to a point, so that $g' = p$, $K$ an arbitrary complex on $X$, and $L$ the constant sheaf, implies that $p_* p^* K = K \otimes f^* g_* \mathbb Q_\ell$ as soon as $K$ is strongly locally acyclic relative to $f$, which it is by Lemma 9.3.4.

Then $g_* \mathbb Q_\ell$ is the cohomology of the affine line, which is simply $\mathbb Q_\ell$, so $K \otimes f^* g_* \mathbb Q_\ell = K \otimes f^* \mathbb Q_\ell = K \otimes \mathbb Q_\ell= K$ (or the same with a different set of coefficients.

Proof of stability: As Deligne notes in the very last paragraph, in the etale setting $p_\#$ is $p_!$ up to a shift and twist: For a smooth morphism, $p^*$ and $p^!$ agree up to a shift and twist, and $p^!$ has a left adjoint $p_!$, which thus agrees with $p_\#$ up to the dual shift and twist.

Since shifting and twisting are equivalences of categories, it suffices to check that $p_! s_*$ is an equivalence of categories. Now $s_*$ of any sheaf is compactly supported over the base $X$ (since a section is compact), which means $p_! s_* = p_* s_*$, and $p_* s_*$ is the identity, and thus an equivalence, by the Leray spectral sequence.

Not sure on the motivation.