Can you source that claim about Iranian soldiers in Iraq? You don't mean the consular officials that were there on the invitation of the Iraqi government do you?
http://en.wikipedia.org/wiki/US_attack_on_Iranian_liaison_office_in_Arbil
About the other thing...I assume that you are talking about al-Sadr and his Mahdi Army. Do you really think that the Shia leaders (such as al-Sadr and al-Sistani) and militias are the insurgency? I think you'll find that they are, relatively speaking, forces of moderation in Iraq. This is in their interests because the American invasion has overthrown the Sunni minority that repressively ruled the country and has handed control of the country to the Shia majority. Al-Sadr wants the Americans out so the Shia can get on with the business of consolidating their power. This is also in Iran's interests. Al-Sistani and al-Sadr are influenced by Iran, but their actions have been primarily political and rhetorical (Najaf being the obvious exception). In fact, al-Sistani has consistently stepped in to broker peace and call for calm whenever tensions have gotten out of hand.
To put it simply, Iranian backed Shia factions in Iraq resist American occupation primarily through political action. The active insurgency consists mostly of the displaced Sunni minority.
Invading Iraq has ultimately served Iran's interests, not America's.