Problem 3  (K&K Ch. 7, problem 1).

For the 1D case, if your answer is too low by a factor of 2, read this:
When you count the number of states with wavevector between k and (k+dk),
you are only concerned with the magnitude of k, not its direction.
So you should also include the states between -k and -(k+dk).
That means you need an extra factor of 2 when you use the density
of modes in k-space, L/(2pi).
If you compare with the 3D and 2D pictures, this will be more clear.
 

Problem 5  (K&K Ch. 7, problem 12).

Remember, you have a formula for <N> for a single orbital.
It is called the distribution function, and we denote it by f(e).
Use equation 5.83, substituting f(e) for <N>.
 

Problem 6  (K&K Ch. 7, problem 3).

Calculate the pressure due to a single orbital first.  Then plug that into
the integral over all orbitals, Eq. 7.12.