"Didn't America commit a major mistake by letting Iran fill up the strategic void left by Iraq?"

No.

"One of the basis strategic principles is to avoid a country becoming a dominant power in a given region."

Known as spheres of influences. It's something Kissenger and Europeans believe strongly in. It's NOT something which Americans generally agree with.
It's idea like Socialism which has failed repetitively but that doesn't interfere with the strong belief in it.

"And one of the consequences of the Iraq war is to have let Iraq open to a strong Iranian political influence."

There is several things wrong with this idea. First there is no reason to assume it's a one way street- why should we suppose that Iran is only going to affect Iraq? Why wouldn't Iraqis influence Iranians. Second any dialogue is usually a good thing. Third the Iranians who going to have most profound affect on people are probably going to be Iranian intellectuals. Iranian Intellectuals could loosely be described as mostly Pro Western [Progress/Liberal].

"More, while one of the war's objectives was to make countries respect the non proliferation treaty, it seems it had the opposite effect persuading Iran to accelerate its nuclear program in order to dissuade the US from attacking it."

If Iran was not developing a nuclear weapons there would no reason for any possible US miltary action. And having nuclear weapons is no deterrance to the US from taking Miltary action against it. [You need proof? We know China has nuclear weapons and this doesn't stop America from planning to defend Tiawan if China attacks it. What is really means is the miltary plans will include this element. And what that means is there has to have [as least one option] the capablity to strike with overwhelming power. So if China actually wanted to invade Tiawan they would in better position [far safer position and more likely to miltarily succeed] if they had no WMD [Nuclear/Biological/or Chemical]. The chance that US would use nuclear weapons against a China with no WMD is practically nil. Or look at Soviet Union did their VAST number of nuclear weapons prevent the US from defending Europe?
Now, having nuclear weapon *might* stop US from some kind of pin prick type of attack, but these aren't very useful anyhow.] A Country having nuclear weapons is does not dissuade the US from attacking it. Whereas a country following internation law does. But it certainly is good argument for Iraqis to decide become a Member of NATO.