Problem 2: Use unit area to get the volume. Make an assumption pressure change.
Problem 3: You need to do an integration over z. Look again at the part about the isothermal atmosphere for inspiration.
Problem 4: Think about what equilibrium temperature means. Start out arguing from an initial temperature to the final one that you have at equilibrium. What do you do to get there? How does this “doing” affect the entropy of the system? You can solve the problem by argumentation with the first and second laws of TD. I am not asking for a mathematical prove here (despite it is possible to do it).