Q3: I think the exam is badly written. But my guess is that you were meant to get the "temperature dependence" in both cases meaning you don't need to worry about numerical prefactors -- so you don't need to do the full integrals. Similar with Debye model.
Q4: It means chemical potential. This unfortunate usage of the word "fermi energy" is common in the semiconductor and engineering community but not common among physicists.