The more compression the more heat. Each compressor stage adds. So yes it's cold enough in the front stages to use composites although the dirt in the air will wear them a lot faster than steel or Titanium. In the high performance engine the compression ratios can drive the temperature at the compressor exit well above 1000 deg F.
For turbines, the more stages the lower the exit temperature, but then there is the efficiency and weight trades and the need to decide whether you want jet pipe thrust or Fan thrust etc. etc.
Somehow I think that the free turbines used in ships (cue Barit1) would have the lowest turbine exit temperature