Temperature measurement
Do the particles of thermometer's bulb attain the same translational, rotational and vibrational energy (vibration of atoms within a particle) as of the particles of the liquid?


No, but they will have the same temperature. Solids don't have any translational energy, the atoms time-averaged positions are constant. If you want to link the temperature to the energy of the particle, it is easiest to start out with a monoatomic ideal gas because here, there is a linear relationship.

Relationship between change in temperature and energy

Although under some ideal circumstances, particles have an energy of $\frac{1}{2} k T$ per degree of freedom, how much energy is needed to raise the temperature of a sample depends on the sample (and the temperature). The heat capacity is a measure of change in energy per change in temperature. So the heat capacity tells you the total energy going into all degrees of freedom when the temperature increases. The splitting up into translational, rotational and vibrational energy depends on the nature of the sample (for example, the physical state) and the temperature.