As far as local max's, etc. A function f(x,y) only has a local maximum or local minimum when the partial derivative in all directions is zero. A saddle point is where the function has a local maximum in one direction (say the x) and a local minimum in another (say the y), where the derivatives are taken at the same point.
Also, your critical points are solved correctly (though for fx = 0 the solution is more generally y = n , where n is an integer and a similar comment holds for your fy solution.) And again, being a bit picky but with reason, we should probably label ALL the coordinates to label the critical point, just to be clear. So your first critical "point" would be (x, 0, 0) (so it's really the whole x-axis) and the second would be (x, , ).
Since the fx critical "point" is not a location where fy is zero the x-axis is neither a local max nor min for the function. The same comment holds for the second critical point. (And thus neither are they saddle points.)