2001 Impala

As for the A/C compressor, first check the fuse/relay. If that's good, your system may be low. The system is designed not to engage the compressor clutch when the system pressure drops below a certain point. You'll have to do a leak test to find out. You could try using a can of R134a that also contains a sealer and dye. The sealer will only be a temporary fix if at all. But the dye will make it easier to do a leak test on the system. If you want to do that yourself, let me know and I'll give you the procedure.

As for the temp gauge, are you saying it doesn't move at all, or just a little? If not at all, check fuse first, then temp sensor on engine, including the wires/connectors. Keep in mind that these gauges sometimes fail themselves and need replacing. Nothing last forever.

If the temp gauge does move but only a little, could be your thermostat is stuck open, or the temp sensor/wiring/connector is bad.