It is conservative, it depends on the path, although the dependence of the force with $x$ is different just because the configuration is different. in case (3) the horizontal force is $F=-kx\sqrt{H^2+x^2}/H$, if H is the height of the hinge and the string is always straight (we also assume that the mass does not move along $y$, also assumed was rest lenght=0).
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