Q1: there is no physical model that gives this equation for arbitrary $m$; the values $m=1,2,3$ appear in viscous flow, as summarized in Viscous Thin Films: For the no-slip boundary condition one has $m=3$, different slip-boundary conditions give $m=2$ (Navier slip) or $m=1$ (Hele-Shaw cell).
Here is a way to understand from dimensional arguments the physics condition $m\leq 3$. Since $u$ has the dimension of length, the gradient $\nabla$ has dimension of 1/length, and the Laplacian $\Delta$ has dimension of 1/length$^2$, the thin-film equation in physical units has the form $$\frac{1}{v_0}\frac{\partial}{\partial t}u=-\lambda^{3-m}\nabla\cdot(u^m\nabla\Delta u).$$ The parameter $v_0$ is a characteristic velocity of the interface, while the parameter $\lambda$ is the slip length at the interface. The no-slip boundary condition has $\lambda=0$, which enforces $m=3$. When there is slip we may have $m<3$, but not $m>3$, because then the right-hand-side would diverge in the limit $\lambda\rightarrow 0$, which is unphysical. (It would imply a divergent interface velocity.)
Q2: this tutorial by J. Hulshof seems a good entry point to the mathematical literature (which considers arbitrary $m$).