do headers and fresh air intakes really work for H/P?

You wouldn't be able to measure any difference.

A cat-back exhaust is best for any bolt on horse power gain. Your motor came with a perfectly good CAI. The after market ones with the cleanable air filters ruin MAF sensors. the oil on the air filter migrates to the MAF sensor wires and ruins them. Headers cause a loss of low end torque off the line and you need a more aggressive cam when running headers.

Headers can allow power increase or just change the power curve. Fresh air intakes are just decoration.

Not enough to make it worth paying a whole bunch of money for them. Those things are WAY over priced.

Headers give you a better sound if you got a V8. For a 4 banger, it just still sounds like a NOISY 4 banger. It helps getting the exhaust out.

Installing an aftermarket CAI on a stock late model fuel injected car is a complete waste money . See this post below for a full explanation

Headers a little, but way up in the RPM range where it is not very usable unless you are drag racing and keep the engine at wide open throttle. Loss of low end torque could also be the result of opening up the exhaust. SO let's say not worth the money on a daily driver.

Cold air intake = complete waste of money. Headers = some improvement but depending on the engine in question the increase in hp is typically marginal and not worth the cost without doing other mods 

Headers do, fresh air intakes, not very much.