There are multiple questions here, but let's start with the basics. You have written two hyperbolic PDEs; (1) is the continuity equation, which is conservative and (2) is the color equation, which is not conservative.

For (1), you have stated correctly that the characteristic speed is $u(x,t)$. For (2), you have made incorrect suggestions in both your answer and your comment. The characteristic speed for (2) is $a(x,t)$. Therefore, the "virtual wind speed" you introduce is not useful.

Yes. Furthermore, and contrary to what @Jan has suggested, (2) is well-posed even for typical discontinuous functions $a$. An upwind discretization for the problem with discontinuous $a$ will be convergent, under the usual restriction on the CFL number.

It is not directly related to your question, but since @Jan brought it up I will add that integrability of $a$ is not the relevant condition. The simplest situation in which a classical solution fails to exist is when there is a point where $a$ changes sign from positive to negative. A delta function must form at that point. Exactly the same thing happens for (1) in the case that $u$ changes sign, so this is nothing special about the color equation.

It depends on what exactly you mean by "upwind methods". For the diffusion equation, there is no directional bias in the transmission of information. An explicit, purely one-sided stencil cannot be convergent for diffusion (see the CFL paper). You could use a stencil that is biased toward one direction (but includes some points in both directions) for any PDE.

For a lengthy and excellent discussion of differences between (1) and (2) and their numerical discretization by upwind methods, see Chapter 9 of LeVeque's book on finite volume methods.