In thermodynamics, you will find different "kinds" of energy. On of them is the one you stated above. There is also the Gibbs energy or Helmholtz energy for instance.
Depending on the system you are studying, you will use a different type of energy. T S P V N are all thermodynamic variables but you won't get anything if they are all allowed to vary. You need some control on your system. In the case were T P and are held fixed, you will use the energy as defined above.
1) How is the differential form of a function obtained? Is it simply the sum of the partial derivatives with respect to each variable multiplied by the differential of the variable?
Just like you said.
What are the advantages of using such an equation instead of the equation of the function itself? You can read off relevant quantities like T P and by varying S V and N respectively. It makes explicits the variables. It makes it easy to switch from one kind of energy to another using Legendre transforms. Probably later you will be ask to derive useful relations from there.