How to find center of ellipse from two points(these are just points on the ellipse, not related to foci), and two radiuses (rx and ry, from standard definition of the ellipse x^2/rx^2 + y^2/ry^2=1) of that ellipse? (there will be two centers, I assume)
You seem to be asking about an ellipse whose axes are aligned with the $x$ and $y$ directions. Note that an ellipse which is rotated by an arbitrary angle is still an ellipse. But you cannot uniquely determine such an ellipse from the information given.
If two points $(x_1, y_1)$ and $(x_2, y_2)$ pass through an ellipse centred at $(x_0, y_0)$ with semi-axes along $x$ and $y$ of length $r_x$ and $r_y$ respectively, then the points $(x_1/r_x, y_1/r_y)$ and $(x_2/r_x, y_2/r_y)$ pass through a circle of unit radius with centre $(x_0/r_x, y_0/r_y)$. So if you can find the centre of this circle, you can find the centre of the corresponding ellipse.