Is there a solid where all triangles on the surface are isosceles? Are there any solids in $R^{3}$ for which, for any 3 points chosen on the surface, at least two of the lengths of the shortest curves which can be drawn on the surface to connect pairs of them are equal? 

There can be no smooth surface with this property, because a smooth surface looks locally like a plane, and the plane allows non-isosceles triangles.
As for non-smooth surfaces embedded in $\mathbb R^3$ -- which would need to be everywhere non-smooth for this purpose -- it is not clear to me that there is even a good general definition of curve length that would allow us to speak of "shortest curves".
