Lennard-Jones potential parameters should not depend on temperature, pressure, etc. And therefore for them it is exact. It is nevertheless always used as approximation for other atoms. Unlike attraction, the exact functional form for the repulsive interaction is not well known and is approximated either by the exponential, or by the inverse power term. Inverse twelfth power is practical for the computer simulations, as it is the square of 6, and hence the energy can be calculated faster.
Minimum energy is at , and is the minimal value of the energy.
The values of this parameters are usually extracted from some experimental data, like crystal structures, thermodynamic properties, kinetic coefficients, etc. They are very hard to get from the first principles.