There are many reasons why Iran does not want an iraqi civil war, partition, or an unstable Iraq.
First, because of the Kurds. Iran has a large Kurdish minority. If a Kurdistan ever came into existence, the Iranian Kurds would start making a lot of noise and trouble to join that state. So would the Turkish kurds. It would reignite the whole Kurdish question, which is the last thing Iran wants.
Second, Iran also has a Sunni minority. If a civil war happened between Sunnis and Shias, it would spill over into Iran. Which, again, is the last thing Iran wants.
Third, despite all the crap you read, an unstable Middle-East is bad for Iran's government. Iran wants stability and economic prosperity. Mahmoud was elected, not very democratically, but still elected by Iranians. If Iran's economy gets too fUcked up, if the civil war spills over into Iran, it's likely he'll loose the next election.
Finally, Iranians are not Arabs. They are Persians. To you it might not matter, but to the Arabs, and the Persians, it's a huge difference. Most of the Arab states around Iran don't like them because of this. Or at least don't trust them because of this. Most of the Arab state certainly do not want a nuclear Iran.
All of this is leverage power for the US, and much more powerful than threats, sanctions, or /ignore.
All these countries are united in their desire to see the US out of Iraq. But once this is done, they will go back to their own squabbles. Iran and Syria are not natural allies. At all. Syria is Baathist, which is basically Arab nationalism. Iran is a Persian Islamist theocracy, which is the complete opposite. They are only circumstantial friends. This is what the US should be playing on, instead of sticking them into the same "evil people" drawer.