Is this an acceptable use of $a \,dx = v \,dv$?

• Are there situations in which an approach like this wouldn't work? (I realize that I assumed constant acceleration when calculating $\bar v$, but anything beyond that?)

Once you start working in more than one dimension, things will get more complicated. In that case, you have $$\mathbf{a} \cdot d \mathbf{x} = \mathbf{v} \cdot d \mathbf{v}$$ and you'll need to keep track of the vector directions. Also, in some cases, the averages you define will not be convenient to compute (as discussed below).

• Is it correct (from a physics standpoint) to say that $a\,dx = v\,dv$ implies $\bar a \Delta x = \bar v \Delta v$? I would prove this mathematically by integrating on both sides, noting that the average value of a function $f(x)$ on an interval $(x_0, x_1)$ is $$\bar f(x) = \frac{1}{x_1 - x_0}\int_{x_0}^{x_1} f(x) dx$$

Sure, but you need to be careful with how you define these averages. You've defined $$\bar{a} = \frac{1}{\Delta x} \int a \, dx, \quad \bar{v} = \frac{1}{\Delta v} \int v \, dv.$$ In other words your $\bar{a}$ is an average over changes in $x$, while your $\bar{v}$ is an average over changes in $v$. But you could also define other averages, like $$\bar{a}' = \frac{1}{\Delta t} \int a \, dt, \quad \bar{a}'' = \frac{1}{\Delta v} \int a \, dv, \quad \bar{a}''' = \frac{1}{\Delta a} \int a \, da, \quad \ldots.$$ So if you forget what kinds of averages you're using, you'll get the wrong answer. You got lucky here because the acceleration was a constant, and any average of a constant is the same.

• Does my work amount to a derivation of the work-energy theorem, or a lucky circumvention of it?

The only essential step you need to prove the work-energy theorem is the chain rule, as I explain in detail here. Using the chain rule to conclude $a \, dx = v \, dv$ and then integrating both sides is the whole derivation of the work-energy theorem. (Though some introductory textbooks somehow make it sound much more complicated...)

• Silly question: If a student did this on a test, would you give them credit?

I would, but in practice it depends on how good your teacher is.