Question (a) has a positive answer in the centrally symmetric case. The proof is involved and I will only summarize the strategy here. Full details can be found in this ArXiv paper. Comments, suggestions, corrections etc are welcome.
Let $K \subset \mathbb{R}^2$ be a compact convex set of area $\pi$. To prove uniqueness of best ellipse approximation, it is sufficient to consider ellipses in the family $E_t := \{(x,y) \in \mathbb{R}^2 ; e^t x^2 + e^{-t} y^2 \le 1 \}$ (because any pair of ellipses can be put inside the family by applying a suitable area-preserving linear map). I actually show that the function $A(t) := \mathrm{area} (K \vartriangle E_t)$ has a unique minimum, and is strictly decreasing (resp. increasing) to the left (resp. right) of it. Equivalently, the function is "strictly quasiconvex".
First, I consider where $K$ is good: this means that that $K$ has a smooth boundary which is transverse to all ellipses $\partial E_t$, except for a finite number of non-degenerate (i.e. quadratic) tangencies, all of which occur outside the envelope hyperbola $xy = \pm 1/2$. In this good case, I prove the following more precise "uniform quasiconvexity" properties:
The function $A$ is $C^2$ except at the tangency parameters.
If $t$ is not a tangency parameter, then: $$ \min\{|A'(t)|, A''(t)\} > \delta(A(t)) > 0, \tag{$\star$} $$ where $\delta$ is some positive continuous function on the interval $(0,2\pi)$ that does not depend on $K$.
If $t$ is a tangency parameter, then the left and right derivatives of $A$ at $t$ exist in $[-\infty,+\infty]$, at least one of them is $\pm \infty$, and $A'_-(t) < A'_+(t)$.
It's not difficult to convince oneself that these properties imply the strict quasiconvexity of the function $A$. Actually any uniform limit of functions with the properties above is also strictly quasiconvex.
Finally, an arbitrary $K$ can always be perturbed with respect to the symmetric difference metric so that it becomes good. So it follows that the function $A$ is always strictly quasi-convex.
How are the quasiconvexity properties proved (in the good case)?
By applying a suitable linear map, it is sufficient to consider $t=0$. Suppose this is not a tangency parameter (as the case of tangency is actually easier). Similarly to my older "answer", one can deduce formulas for the derivatives $A'(0)$ and $A''(0)$. Actually $A'(0)$ depends only on the points of intersection between $\partial K$ and the unit circle $\partial E_0$, while $A''(0)$ depends also on the angles of intersection. An inspection of the second formula shows that $A''(0)$ has a strong "tendency" for being positive; for example if no two consecutive intersection points are separated by an angle $>\pi/2$ then $A''(0)>0$. What we need to show in order to obtain the main inequality $(\star)$ is that if $A''(0) \leq 0$ (and $A(0)$ is not too close to $0$ or $2\pi$) then $A'(0)$ cannot be too close to zero. This is done by a case-by-case analysis, combining analytic and geometric arguments -- I can only refer to the paper for the details.