I have measured a bunch of intake air temps. The intercooler indeed works to cool the air charge (especially for high-boost setups, but still helps noticeably at stock boost pressure), assuming the intercooler setup is free-flowing and efficient (a measure of how much degrees temperature it cools compared to how much it possibly could if it were a "perfect" heat exchanger - IE, 100% intercooler would have air coming out of it the same temperature as ambient air temps), it is definitely beneficial. The result is more air density which allows for noticeably improved power with adequate fueling.
Back when I did the air intake temp measurements, I also took some measurements inside the intake manifold as well. Indeed, when the engine was switched off so air was stagnant in the intake manifold, the air inside there would rise in temperature dramatically due to the heat soak. The air cooled down dramatically after it was running the the car in motion to get airflow through the engine compartment cooling it off. A quick way to measure the temperature of the intake manifold is with your hand - give it the test of whether it's too hot to touch or not. That works for me in a pinch when I don't have my pyrometer thermocouple probe setup all handy. I found that between autocross runs, I can keep coolest intake manifold temps by keeping the engine idling constantly with the hood closed. That keeps the intake manifold cool enough to touch. Engine off with hood open, really heats up the intake manifold smokin' hot. Not good for autocross!
Ceramic coating on the inside and outside of the exhaust manifold helps the turbo to spool up a little bit quicker. In theory, it might help reduce intake manifold temps a little, but fact is the exhaust manifold still gets HOT even with some coating on it.