  Let $\pi'$ be a fixed unitary cuspidal representation of
$\mathrm{GL}(n)/\mathbb{Q}.$ We establish a subconvex bound in the $t$-aspect
  $$
L(1/2+it,\pi\times\pi')\ll_{\pi,\pi',\varepsilon}(1+|t|)^{\frac{n(n+1)}{4}-\frac{1}{4\cdot
(4n^2+2n-1)}+\varepsilon}, $$ for any unitary pure isobaric automorphic
representation $\pi$ of $\mathrm{GL}(n+1)/\mathbb{Q}.$ Moreover, the bound
improves in the standard $L$-function case $$ L(1/2+it,
\pi')\ll_{\pi',\varepsilon}(1+|t|)^{\frac{n}{4}-\frac{1}{4(n+1)(4n-1)}+\varepsilon}.
$$
  We also prove an explicit lower bound for nonvanishing of central $L$-values
$$ \sum_{\pi\in\mathcal{A}_0}\textbf{1}_{L(1/2,\pi\times\pi')\neq
0}\gg_{\varepsilon}|\mathcal{A}_0|^{\frac{1}{n(n+1)(4n^2+2n-1)}-\varepsilon},
$$ for a suitable finite family $\mathcal{A}_0$ of unitary cuspidal
representations of $\mathrm{GL}(n+1)/\mathbb{Q}.$ More generally, we address
the spectral side subconvexity in the case of uniform parameter growth, and a
quantitative form of simultaneous nonvanishing of central $L$-values for
$\mathrm{GL}(n+1)\times\mathrm{GL}(n)$ (over $\mathbb{Q}$) in both level and
eigenvalue aspects.
  Among other ingredients, our proofs employ a new relative trace formula in
conjunction with P. Nelson's construction of archimedean test functions in
\cite{Nel21} and volume estimates in \cite{Nel20}.
