How does the outflow valve work on a cabin pressurization system?
Pressure does not rise because keeping the cabin pressurized begins with releasing air to the outside. The flight starts low, with low cabin altitude, as the plane climbs the cabin altitude rises (lower pressure than it started).

As for the air itself. When it is compressed by the ACM, the temperature does rise, that's why it is cooled before being introduced into the cabin.

See here: Where does the final cooling take place in the air cycle machine of the air conditioning system?

Work is done on the air, then the air does work by passing through a heat exchanger, and then a turbine, which runs the ACM.

There is one plane I know of that maintains sea level pressure up to FL 410. In that case air will still be jettisoned overboard to bring in fresh air, with the same ACM cycle of temperature regulation.