Differentiating $m(t)$ and using the equation leads to
$$
m'(t):=\int_0^{\infty}\left(
\frac{\sigma^2(t)}{2}\partial^2_{xx} p(x,t) - b(t)\partial_x p(x,t)\right)\,dx=
-\frac{\sigma(t)^2}{2}\partial_{x} p(0,t)<0.
$$
The last inequality follows from the Zaremba-Giraud theorem for the sign of the solution's normal derivative at the boundary. See, for example, Nazarov A.I., A Centennial of the Zaremba–Hopf–Oleinik Lemma.
For $b\equiv0$
$$
p(t,x)=\int_0^{\infty}\frac{1}{\sqrt{2\pi \int_0^t{\sigma^2(\tau)}\,d\tau}}\left(\exp\left(-\frac{(x-y)^2}{2\int_0^t{\sigma^2(\tau)}\,d\tau}\right)-\exp\left(-\frac{(x+y)^2}{2\int_0^t{\sigma^2(\tau)}\,d\tau}\right)\right)\rho(y)dy.
$$
For the general case the Green's function of the first BVP cannot be expressed via elementary functions.
One has the ordering of the three $\lambda$'s, i.e.
$$
\lambda_{\text{convex}} \leq \lambda_{\text{LSI}} \leq \lambda_{\text{SG}},
$$
where $\lambda_{\text{convex}}$ is the one from convexity, $\lambda_{\text{LSI}}$ is the one from the inverse Log-Sobolev constant (both as in your question) and $\lambda_{\text{SG}}$ the spectral gap, characterized via the smallest constant in the weighted Poincaré inequality
$$
\Vert f -1 \Vert_{L^2(\pi)}^2 \leq (2\lambda)^{-1} \int |\nabla f |^2 d\pi , \qquad \forall f : \int f \, d\pi = 1.
$$
here $f$ plays the role of $d\rho/ d\pi$. The first inequality is the HWI-inequality after Otto-Villani and the second is linearization of the LSI.
The equivalence $\lambda_{\text{convex}} = \lambda_{\text{LSI}} = \lambda_{\text{SG}}$ holds for $V$ being a positive quadratic form, i.e. $V(x) = x \cdot H x$ for a fixed symmetric positive matrix $H$. Then, the eigenvalues and eigenvectors of the according Ornstein-Uhlenbeck process are explicit (products of Hermite-polynomials) and obtained in dependence of the eigenvalues of $H$. The smallest non-negative eigenvalue is then the smallest eigenvalue of $H$, in accordance to the $\lambda$-convexity. The LSI is sandwiched inbetween anyways.
Except for this particular case, I expect that the equivalence breaks down for generic $V$ (non-quadratic).
This is easiest observed in the non-convex case. So let, $V$ be a double-well, with local max in $0$ and two minima in $~\pm 1$ and have convex quadratic growth outside a bounded region. Then, the Fokker-Planck evolution is $\lambda$-convex for a $\lambda<0$ being the lowest bound on the Hessian again, determined by the non-convexity of $V$ around the local maximum. However, the Log-Sobolev and spectral gap constants are still positive finite and can be obtained by combining the Bakry-Emery criterion with the Holley-Stroock perturbation principle, since $V$ is a bounded perturbation of a convex potential.
This becomes even more apparent, by considering the vanishing diffusion limit, i.e. consider for $\varepsilon>0$ the Fokker-Planck equation
$$
\partial_t \rho = \varepsilon \Delta \rho + \nabla \cdot(\rho \nabla V)
$$
In this case, it becomes clear that the time-scales captured by convexity and Log-Sobolev constants or spectral gaps are rather different.
From the comment of @Tobsn follows that, convexity measures local stability at every point in the space of probability measures and the setting of double-well potential it is readily checked that, one actually gets the opposite comparison of the type
$$
W(\rho_t, \hat\rho_t) \geq c e^{|\lambda| t} W(\delta_{-\eta},\delta_{\eta}) \quad\text{for } t \in [0, t_0]
$$
where $\eta>0$ is small and $t_0$ is also not too large. I write $|\lambda|=-\lambda$ to make clear, that the trajectories expand and do not converge.
This follows, because $\rho$ and $\hat\rho$ will follow mainly the deterministic ODE $\dot X_t = - \nabla V(X_t)$, which expands at rate $|\lambda|$ close to the local maximum. In particular, this result also holds for $\varepsilon=0$.
In comparison, the log-Sobolev constant and the inverse spectral gap measure a global time-scale quantifying the ergodicity in time, showing that the diffusion converges to the measure $\pi$ in entropy or the relative density $f_t = \rho_t / \pi$ in the $L^2(\pi)$-sense. Those are global averaged quantities and hence behave in general better, as the non-convex setting shows. However, note that for the $\varepsilon$-dependent case, both $\lambda_{LSI}$ and $\lambda_{SG}$ will degenerate to $0$ exponentially like $e^{-C/\varepsilon}$, which is a statement about metastability being present in this setting (the limiting ODE dynamic is non-ergodic, since it has actually three stationary states).
There is a theory of variable Ricci bounds, which can catch a bit this different local stability and might improve the gap between those constants. One, can probably think of the Bracamp-Lieb inequality as an instance of variable curvature, since it shows that $\text{Hess}\, V$ behaves like the local Ricci-tensor for the diffusion, that is
$$
\Vert f - 1 \Vert_\pi^2 \leq \int \left\langle \nabla f , \text{Hess} V\, \nabla f \right\rangle d\pi .
$$
Other ways, to make the different behaviour of the constants precise, at least in the scaling regime with vanishing diffusion $\varepsilon\ll 1$, are in 1d upper and lower estimates on the Log-Sobolev constant and spectral gap with the help of the Bobkov-Götze or Muckenhoupt criterion.
I can provide more details or references on the individual mentioned observations, but I'm not aware of a general result showing that for generic V (not perfectly quadratic potentials), there are strict inequalities between all of the three $\lambda$'s, i.e.
$$
\lambda_{\text{convex}} < \lambda_{\text{LSI}} < \lambda_{\text{SG}}.
$$
Best Answer
Let me rewrite the equation (3) as $$ \partial_t \rho = \partial_x \left(\rho\, \partial_x\log\left(\frac{\rho}{\mathcal{F}[\rho]}\right)\right) \label{3'}\tag{3'}. $$ Then, this equation is a gradient flow with respect to the metric tensor after Otto inducing the Wasserstein distance iff there exists a driving free energy $\mathcal{E} : \mathcal{P}(\mathbb{R}^d) \to \mathbb{R}$ such that its variational derivative satisfies $$ \mathcal{E}'(\rho) = \log\left(\frac{\rho}{\mathcal{F}[\rho]}\right) + C, $$ where the constant $C$ does not matter and could even depend on $\rho$.
In the first case with $\mathcal{F}[\rho]\equiv \rho_\infty$, one gets $$ \mathcal{E}(\rho) = H_{\phi}[\rho] \quad\text{with}\quad \phi(r) = r \log r -r + 1 . $$ A more interesting case is for some potential energy $V:\mathbb{R}^d \to \mathbb{R}$ and symmetric interaction energy $W:\mathbb{R^d}\times\mathbb{R^d} \to \mathbb{R}$ the map $$ \mathcal{F}[\rho](x) = Z[\rho]^{-1} \exp\left( -V(x)- \int W(x,y) \rho(y) dy \right) \quad\text{with}\quad Z[\rho] := \int \exp\left(- V(x)- \int W(x,y) \rho(y) dy \right) dx .$$ Then, upto a $\rho$-dependent constant, the free energy is given by $$ \mathcal{E}(\rho) = \int \rho \log \rho\, dx + \int V(x) \rho(x) \, dx + \frac{1}{2} \int \int W(x,y) \rho(x)\rho(y)\, dx\, dy . $$ This is the classic McKean-Vlasov model and the free energy above consisting of entropy, potential energy and interaction energy is studied a lot in different areas (gradient flows, density functional theory, statistical mechanics).
Depending on your map $\mathcal{F}$ there might be other free energies. The above one is the most common in the literature of mean-field limits for interacting particle systems, where it is usually even assumed that $W(x,y)=w(x-y)$ for some even $w:\mathbb{R}^d \to \mathbb{R}$.