Equations of the ellipse
If you have a rectangle with center $(h,k),$ width $2a$ parallel to the $x$ axis, and height $2b$ parallel to the $y$ axis, then the equation of the ellipse is
$$ \frac{(x - h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1. \tag1$$
That is, a point is on the ellipse if it satisfies Equation $(1)$.
The condition for a point to be inside the ellipse is
$$ \frac{(x - h)^2}{a^2} + \frac{(y-k)^2}{b^2} < 1. \tag2$$
And of course a point is on or inside the ellipse if it satisfies
$$ \frac{(x - h)^2}{a^2} + \frac{(y-k)^2}{b^2} \leq 1. \tag3$$
There is also a set of parametric equations for the ellipse:
\begin{align}
x &= a \cos \phi \\
y &= b \sin \phi,
\end{align}
which look somewhat like the parametric equations for the unit circle, except that we have introduced new factors ($a$ and $b$) and the parameter $\phi$ is not the angular direction to the point $(x,y)$ from the origin.
Instead, if the angular direction to $(x,y)$ is $\theta,$ then
$$ a \tan\theta = b \tan\phi, $$
which enables conversion of any angle $\theta$ to the parameter $\phi$ and vice versa.
Finding the endpoints of the arc
Let $\theta$ be the angular direction from the origin to one endpoint of the arc (either the start or the end, take your pick).
There are at least two ways to get the coordinates $(x,y)$ of this endpoint. One is to solve a system of two equations: Equation $(1)$ together with the equation
$$ (x - h) \sin\theta = (y - k) \cos\theta. $$
Depending on $\theta,$ this gives either $x=0,$ $y=0,$ or $y$ as a linear function of $x$; you can make this substitution into Equation $(1)$ and solve using the quadratic formula. You will get two answers; choose the one in which the signs of $x$ and $y$ agree with the signs of $\cos\theta$ and $\sin\theta,$ respectively.
Another way to find the coordinates $(x,y)$ of the endpoint is to set
$$ \phi = \mathrm{atan2}(a \sin\theta, b \cos\theta) $$
where $\mathrm{atan2}(p,q)$ is the function that gives $\pm\arctan\frac pq$ with the sign chosen so that the sine and cosine of the result have the same signs as $p$ and $q,$ respectively. In other words, $\mathrm{atan2}(p,q)$ gives the direction to $(q,p).$ If you must use the ordinary arc tangent, then take $\phi = \pm\arctan\left(\frac{a}{b} \tan\theta\right),$
choosing the sign to be the same as the sign of $\cos\theta.$
Testing if $(x,y)$ is inside the segment of the arc
Find the endpoints of the segment. Find the equation of the line through those two points. Find an inequality satisfied by all points on the same side of that line as the arc. (There are several ways to do this.)
The points in the segment are the points that satisfy the inequality in the previous paragraph and also satisfy Inequality $(2)$ above.
Testing if $(x,y)$ is inside the sector of the arc
For starting angle $\theta_1$ and ending angle $\theta_2,$
write an inequality for being on the "correct" side of the line
$y = k + (x - h) \tan\theta_1$
and one for being on the "correct" side of the line
$y = k + (x - h) \tan\theta_2.$
In the example in the question, the selected arc is inside a $30$ degree angle between lines from the center of the ellipse and is less than half of the entire ellipse.
But if the arc goes counterclockwise from a start angle of $20$ degrees to an end angle of $220$ degrees, for example, the arc is more than half of the ellipse; the remaining part of the ellipse that is not part of the arc is smaller.
If the arc is less than half of the ellipse, $(x,y)$ is in the sector only if the inequality for $\theta_1$ is true and the inequality for $\theta_2$ also is true.
If the arc is more than half of the ellipse then the required condition is that at least one of the inequalities is true; it is not necessary for both to be true.
In either case (less than half the ellipse or more than half), $(x,y)$ is inside the sector only if Inequality $(2)$ is true.
Testing if $(x,y)$ is on the arc
This is the same as testing for "inside the segment" or "inside the sector" (your choice), except that instead of using Inequality $(2)$ you use Equation $(1)$.
Testing if $(x,y)$ is on the segment or sector border
In addition to the arc, the boundary of the segment consists of the line segment between the endpoints of the arc, and the boundary of the sector consists of the two line segments between $(h,k)$ and the endpoints of the arc.
There are well-known ways to determine whether a point $(x,y)$ is on the line segment between two given points. For the segment of the ellipse, however, and alternative is to check whether $(x,y)$ satisfies the equation of the line through the two endpoints of the arc and also satisfies Inequality $(3)$.
For the sector you can check whether $(x,y)$ is in one of the correct directions from $(h,k)$ (either $\theta_1$ or $\theta_2$) and also satisfies Inequality $(3)$. But this has disadvantages for practical computing, since it is hard to set tolerances on an angle to cover numerical errors in the distance from a line.
Testing if $(x,y)$ is on a thick segment or sector border
In most cases this comes down to whether $(x,y)$ is within half a border's thickness of the nearest curve or line segment bounding the ellipse.
There is an extra complication at the "corners" of the segment or sector;
is the border rounded at those points, mitered, or finished some other way?
Distance from a line is relatively easy to compute, but distance from an ellipse is relatively messy. In general it involves a quartic (fourth-degree) polynomial. See the answers of Distance of point $P$ from an ellipse or Calculating Distance of a Point from an Ellipse Border for some clues.
