Q:

Find the minimum distance to move an ellipse to be inside another ellipse?

For the problem of ellipse intersection,
I would like to know an accurate "general, including the cases of two non intersected ellipses, and non aligned ellipses" method to calculate the minimum Euclidean distance to translate an ellipse to be inside another ellipse, so that:

I have an ellipse "A" and "B"
I want ellipse "A" to be inside ellipse "B"

A:

Write the stationary ellipse $B$ in implicit form: $V(x,y) = 1$ where $V$ is a quadratic polynomial in $x,y$.  Write the ellipse $A$ in parametric form:
$x = f(\theta)$, $y = g(\theta)$, $0 \le \theta \le 2\pi$ where $f$ and $g$ are linear combinations of $\cos(\theta)$, $\sin(\theta)$ and $1$.  Ellipse $A$ translated by $(s,t)$ is $x = s + f(\theta)$, $y = t + g(\theta)$.  To have the translated $A$ inside $B$, you need $V(s+f(\theta), t+g(\theta)) \le 1$ for all $\theta \in [0,2\pi]$.  The optimal solution will have $$V(s+f(\theta_0), t + g(\theta_0)) = 1$$ for (at least) one $ \theta_0$, with $$\left.\dfrac{\partial}{\partial \theta} V(s+f(\theta), t+g(\theta))\right|_{\theta = \theta_0} = 0$$
Eliminating $\theta_0$ from these two equations will give you a polynomial equation $P(s,t) = 0$.  Then you want to minimize $s^2 + t^2$ subject to that.  Use a Lagrange multiplier.
Caution: $P(s,t) = 0$ gives you $(s,t)$ where the two ellipses are tangent.  Not all of these solutions will have the translate of $A$ inside $B$, so you need to examine each local extremum found in the last paragraph.  You also need to look at self-intersections of $P(s,t)=0$, where the two ellipses are tangent at two (or more) points.
Unfortunately, this approach will require high-precision solving of some rather nasty high-degree polynomials.  I don't see any way around that.
For example, I tried the ellipses $x^2/4 + y^2 = 1$ (red) and $x = 2+\cos(\theta)/2, 2 - \cos(\theta)/2 - \sin(\theta)/4$ (blue).

The polynomial $P(s,t)$ turns out to be
$$ 400\,{s}^{8}+1280\,{s}^{7}t+4864\,{s}^{6}{t}^{2}+11264\,{s}^{5}{t}^{3}
+19968\,{s}^{4}{t}^{4}+28672\,{s}^{3}{t}^{5}+28672\,{s}^{2}{t}^{6}+
16384\,s{t}^{7}+4096\,{t}^{8}+8960\,{s}^{7}+37376\,{s}^{6}t+125952\,{s
}^{5}{t}^{2}+272384\,{s}^{4}{t}^{3}+446464\,{s}^{3}{t}^{4}+516096\,{s}
^{2}{t}^{5}+344064\,s{t}^{6}+98304\,{t}^{7}+94072\,{s}^{6}+465472\,{s}
^{5}t+1400096\,{s}^{4}{t}^{2}+2781696\,{s}^{3}{t}^{3}+3785344\,{s}^{2}
{t}^{4}+2958336\,s{t}^{5}+988672\,{t}^{6}+624160\,{s}^{5}+3222272\,{s}
^{4}t+8672512\,{s}^{3}{t}^{2}+14565376\,{s}^{2}{t}^{3}+13627904\,s{t}^
{4}+5394432\,{t}^{5}+2822609\,{s}^{4}+13608256\,{s}^{3}t+31176712\,{s}
^{2}{t}^{2}+36580608\,s{t}^{3}+17556240\,{t}^{4}+8565512\,{s}^{3}+
35372960\,{s}^{2}t+57681440\,s{t}^{2}+35238528\,{t}^{3}+16651594\,{s}^
{2}+49842824\,st+43031000\,{t}^{2}+18303224\,s+29487984\,t+8741466
$$
Here is a plot of $P(s,t) = 0$:

The outer curve has the two ellipses tangent externally.  The closest point to the origin on the inner curve is the cusp at approximately $$s = -.51323143831669984893354538119451, t = -1.4084700967403685126969584735633$$
But that doesn't work: it seems to correspond to a third-order contact, and the blue ellipse is not inside the red one:

The next point to try is the self-intersection at approximately
$$s = -.548586303703105857310127208690, t = -1.69940238118988100188408119644$$
This one does work: the two ellipses are internally tangent at two points.

Rather close to that point, and slightly closer to the origin, there is a local minimum of $s^2 + t^2$ on the curve, approximately
$$ s= -.53385128947026720961128611, t =-1.70397049534824072399428450222
$$
The picture here is very similar to the previous one, but it turns out that the blue ellipse sticks out of the red one slightly in this case: the maximum value of $V$ on the blue ellipse is about $1.0165$.

