Can quantum mechanics model a classical particle in a box? Classically, a particle will be able to exhibit sideways motion as it bounces from one end of the box to the other. Can we form a Gaussian like wavepacket from the stationery state solutions? If so, when these solutions evolve over time, do they cause the position distribution to exhibit 'bouncing'? And what does the momentum space distribution look like in this case? The momentum distribution for a single eigenstate is an even function, but surely with a bouncing particle it should be localised?
How does Q.M correspond to classical predictions in the limit of large L?
 A: The classical limit of the particle in a box would indeed be a wave packet with some range $\Delta n$ of values.  This page discusses this limit in a bit of detail, and provides a handy applet that allows you to play around with various wavepackets and see their evolution in the position and momentum representations.
To summarize in case of link rot:  a superposition of states between $n_0-\Delta n$ and $n_0+\Delta n$, with $\Delta n \ll n_0$, gives a wavepacket of the type you want.  The wavefunction will then be
$$
\Psi(x,t) = \sum_{n=n_0-\Delta n}^{n_0+\Delta n} C_n \sin \left( \frac{n \pi x}{a} \right) e^{-i E_n t/\hbar} 
$$
The coefficients $C_n$ for the $n$th eigenstate are somewhat arbitrary;  we choose them to be
$$
C_n \propto \cos^2 \left[\frac{(n- n_0) \pi}{2 n_0 + 1} \right] e^{-i (n-n_0)\pi/2}
$$
(Simply replacing the cosine above with a constant will yield similar results but the initial wavepacket won't be as smooth.)
This leads to a "smooth" wavepacket that "bounces around" inside the box for some time.  However, it also disperses with time (i.e., $\Delta x$ increases), as we would expect from a free particle.  The wave packet also interferes with itself substantially when it bounces off of the "walls" of the box.

A: The ubiquitous 'particle in an infinite square well' examples are using the time-independent Schroedinger equation to look for stationary states. Because those are very simple to find. Unsurprisingly, they find stationary states, and have no time dependence. That's not really a feature of Quantum Mechanics, just of the simple illustrative example.
It is possible to model non-stationary states where the particle starts with some initial position and momentum functions and observe how they change over time, with the modal position moving around the box.
