Is "more added heat, more molecular vibration" an universal property? Or are there exceptions? Is "more added heat, more molecular vibration" an universal property? Or are there exceptions?
Do some substances work the other way around? Or do some other ambient properties modify this property? Such as if the substance is applied electrical current at the same time? Or e.g. submerged to some other substance?
 A: No. Consider a phase transition. This requires heat, but the temperature stays the same. So classically, the kinetic energy would remain the same.
And then there are spin systems where there is no movement, so it is not universal.
Adding heat $Q$ means adding entropy $S$: ${\rm d}S = \delta Q/T$.
A: In addition to Pieter's answer which claims that during a phase transition adding more heat does not contribute to a change in kinetic energy, I add that in some phase transitions, adding more heat converts a liquid to a solid, thereby lowering the kinetic energy of the molecules.
I am not sure I agree with Pieter's answer, because I'd argue that even though the temperature stays the same during a phase transition, if liquid is converted to gas or solid is converted to either liquid or gas, the kinetic energy of the molecules do increase.
Edit: As requested by Pieter, a concrete example of a liquid material that becomes a solid when heat is added is helium 3. See the corresponding phase diagram: 
Note that it isn't the only substance having this property. I do not remember the exact name of such substances, but I do remember a good thermoelectric one having that property too.
A: For a system of molecules always $T\geq0$. With the standard definition of temperature as a number proportional to the average energy in the degrees of freedom of a system, always $T\geq0$. For more exotic systems, it depends on how you define temperature. You can define temperature as the inverse rate of increase of entropy with energy, that is $T=\left( \frac{dS}{dE}\right)^{-1}$. With this definition negative temperature is possible. Examples are give here.
