First note that the process is an irreversible adiabatic expansion. If it were a reversible one, the gas has to expand against a varying pressure because pressure of gas decreases during expansion and this varying gas pressure must be matched with external pressure in all infinitesimal steps of expansion. "The gas expanded against constant pressure" ⇒ the gas must have a pressure which is always higher than the constant external pressure during the expansion ⇒ irreversible.
Will entropy definitely increase?
ΔSsur = 0 as q = 0. Irreversible ⇒ ΔStot > 0. ΔStot = ΔSsys + ΔSsur = 0 + ΔSsys = ΔSsys. So ΔSsys > 0, so entropy of system will definitely increase.
Will compressibility factor definitely increase?
In adiabatic irreversible expansion all p, V, T changes. If Z is to increase, the increment in pV must be greater than the increment in T, or the magnitude of decrement in T must be greater than that in pV, which cannot be sure will happen as all p, V, T change.
What did you do wrong?
First of all, TV equation is for perfect gas doing reversible adiabatic expansion only. Second of all, a gas expands against constant external pressure does not mean the pressure of gas does not change. All p, V, T will change. You said as volume increases, temperature decreases, but pressure of gas will decrease too.