In the first line of the solution of exercise 2.1, I think there is a factor g(\omega) missing. Also, the integral has limits, but the integration variable is a vector, so that does not make sense.
I also think that in the very last line of the solution of exercise 4, the upper limit in the integral in the formula for C should be infinity since we are in the low-T limit. If not, I think the differentiation would be incorrect since the integral is a function of \beta and therefore of T and should be included in the differentiation.