Question

Do kinetic energy increase during phase change? And do potential and kinetic energy both increase during the rise of temperature?

Answer 1

During a change of phase the temperature does not change, but the internal energy does. The internal energy is the sum of the kinetic energy of the molecules and the chemical potential energy of the molecules.During a change of phase, the average kinetic energy of the molecules stays the same, but the average potential energy changes.
Yes, potential energy increases with increasing temperature for at least the following three reasons:

At a higher temperature, more atoms/molecules are in excited electronic states. Higher electronic states correspond to greater potential energy. Potential Energy is -2 times Kinetic Energy. So actually, at higher temperature, when more atoms are in higher electronic states, there is more potential energy and less kinetic energy (just considering electronic energy).

At higher temperature, more molecules are in excited vibrational states. Higher vibrational states correspond to greater potential energy. Half the energy of each vibrational state is kinetic, half potential.

At higher temperature, more molecules are in excited rotational states. While for a rigid rotor, this would not involve increased potential energy, real molecules are not rigid. The rotation of a diatomic molecule about an axis perpendicular to the bond stretches the bond as a centrifugal distortion, which represents an increase in potential energy.