CompressorSurge I'll try to answer all your questions here.
Yes, density altitude increase reduces thrust, just like piston engines. This applies to hot and high on the ground too, air density is reduced there as well.
Fuel/air ratio is metered by the FCU or FADEC - that's what it's there for. So fuel/air ought to be correct all the time.
Any form of forced induction, supercharging or turbocharging, simply jams more air into the induction system thereby increasing local air density and hence power available. More air = more fuel able to be burnt.
The diffuser case is placed after the last stage of compressor to expand the airmass and allow it to slow down, thereby converting kinetic energy gained through the compressor stages to pressure energy. Normal gas laws apply but be aware of the temperature of this air (hot!).
This process, diffusion, will increase the density of the air processed by the compressor, that's what it's there for. Higher density, lower density altitude, which allows more fuel to be burnt by the air processed and hence more thrust. The compressor and diffuser sections work together to maximise overall compression ratio and hence maximise thrust available.
Gas turbine engine performance, just like all other air breathing engines, varies directly as air density. Increase or decrease air density and increase or decrease thrust. Note however that increasing density altitude means reducing actual density. The reduced air density allows the aircraft to fly producing less drag, requiring less thrust, because there are less air molecules to bump into. Engines can also be designed to turn up more RPM at altitude and gain a bit of thrust. Intake air is colder so air entering the combustion section can stand a bit more fuel burning and stay below TIT limits. There is also ram recovery at the inlet increasing the pressure (hence density) at the burner section entry. Ram recovery is, simply put, that the engine inlet is "pushed" through the air and the air is "shoved" in because it simply can't get out of the way.
You asked a lot with just a few lines old mate!
I hope this helps.