Is a circle classified as an ellipse?

Yes. A circle is a special case of an ellipse.

The equation of an ellipse centered at the origin $(0,0)$ is: $$\left(\frac xa \right)^2 + \left(\frac yb \right)^2=1$$

When $a=b=1$, this gives the equation of the unit circle: $x^2+y^2=1$.

In general, if $a=b=r$, you get the equation of a circle with radius $r$: $$x^2+y^2=r^2$$

Can you come up the equation of a circle not centered at $(0,0)$, but instead centered at $(h,k)$?

Both the foci of a circle coincide and thus, its eccentricity is zero. So yes, it is an ellipse.
It is like that all squares are rectangles but all rectangles are not squares.
See this link- [the most intuitive link ever seen!] http://www.mathsisfun.com/geometry/eccentricity.html