Reference request: Long-term behaviour of the heat equation for bounded initial data

First of all, with initial data in $C_b$, classical solution exist, so there is no need for quotation marks. It is easy to see that the convolution of the initial data $f(x)$ with the Gauss–Weierstrass kernel $p_t(x)$ defines the unique bounded classical solution $u(t, x)$.

It is easy to see that $u(t,x) - u(t,y)$ converges to zero as $t \to \infty$: the functions $p_t(\cdot - x) - p_t(\cdot - y)$ converge in $L^1$ to zero as $t \to \infty$. It is only slightly more difficult to see that this convergence is locally uniform with respect to $x$ and $y$. Therefore, your question is equivalent to the following simpler one: for what initial data $f(x)$, the limit of $u(t, 0)$ exists as $t \to \infty$.

I doubt there is a simpler "if and only if" characterization. One can easily provide sufficient conditions, for example it is sufficient to assume that the mean value of $f$ over the ball $B(0, R)$ has a limit as $R \to \infty$ (by the same argument that one uses when proving that Cesàro convergence implies Abel convergence). And one can equally easily construct counterexamples.

I doubt that there is a characterization, but one thing I can say is that the complement of the set of such $f$ contains a dense open set.

Let $U(f)$ be the solution with initial condition $u(0,x) = f(x)$, and $$G = \{f \in C_b: \lim_{t \to \infty} U(f)(t,\cdot)\ \text{converges uniformly on compact sets}\}$$

Take some $f_0 \in C_b$ such that $U(f_0)(t,0)$ does not converge as $t \to \infty$. Let $\delta = \limsup_{t \to \infty} U(f_0)(t,0) - \liminf_{t \to \infty} U(f_0)(t,0) > 0$. Then for any $f \in G$ and any $c \ne 0$, if $\|g- (f + c f_0)\|_\infty < |c|\delta/2$ then $$\limsup_{t \infty} U(g)(t,0) - \liminf_{t \to \infty} U(g)(t,0) \ge |c|\delta - 2 \|g-(f+c f_0)\|_\infty > 0 $$