I would like to clarify my question. I have numbered them to be independent questions
For any conservative fields, $\vec{F} = -\nabla U$. Which means the restoring force is opposite to the increasing values of the potential energy. If it so happens that $F = \nabla U$, does that mean the restoring force (not even called that anymore I guess) would be in the same direction as the increasing values of energy? Is this a violation to the first law in thermodynamics.
For attracting bodies, the closer the objects come together, the potential energy drops as $r \to 0^-$ and $U \to -\infty$, but for repelling bodies we get $r \to 0^-$ as $U \to \infty$ which means the potential energy actually increase. Am I right?