introducing exhaust gas into the mix is done because exhaust gas is inert (its already burnt so it can't be burnt again). this inert gas tends to 'snuff out' the flame during combustion resulting in lower PEAK combustion temps. the lower temp means nitrogen doesn't get burnt to form Nox (pollution).
yes you may lower the intake temp and therefore increase the oxygen in the mix but this in turn (due to more oxygen) will increase the combustion temperature.....and you have double edge sword contributing to higher engine temp
yes and no. disabling EGR increases PEAK temp but lowers the engine temp.
i didn't quote the bit about specific heat capacity as it makes it confusing and long winded

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you need to remember that intake air has a big cooling effect on the motor. the main source of heat in a diesel is from the fuel. how hot the motor runs depends on how well its cooled. roughly put the more air you shove in the cooler it runs as the air cools the motor.
when you replace that cool air with 400°C exhaust gas the motor will run hotter simply because you removed what was cooling it.
Furthermore I don't understand how you can lower the combustion temp and increase the exhaust temp as a result?
remember that regardless of EGR the overall amount of heat comes from burning the fuel. burn the same amount of fuel you will get the same heat regardless of EGR or peak temps.
an example... EGR off, inlet air temp is 30°C, add fuel and burn. exhaust temp is 600°C.
now with EGR on, inlet temp is 200°C, ad the same amount of fuel and burn, exhaust temp is 200°C + the heat from the fuel(600°C) + more due to lack of cooling from the intake, - a bit due to peak combustion temp is lower, so you get exhaust temp = 700°C.
basically the EGR drops the PEAK temp but increases the AVERAGE temp. its this average temp which overheats motors and melts pistons.
the higher non-egr peak temp is not a problem as the motors actually turn off EGR at full throttle anyway, which is where you also have the hottest engine temps due to maximum fuel.
turbo's throw a small curve ball into the mix. when you go full throttle off boost you have limited amount of air going into the motor but lots of fuel. this rich mix nad limited cooling means high exhaust temps. if you have EGR on and especially if the inlet manifold is sooted up restricting the air flow (which is needed to keep the exhaust temps down) then you can have dangerously high exhaust temps very easily.
if you run an EGT gauge you will probably notice the EGT's drop when you fitted the intercooler. decreasing the inlet temps decreases exhaust temps.