The mosque bombing has Iran written all over it. They've had their hands involved in the "insurgency" in Iraq since the fall of Saddam. Hell, that's where Zarqawi flees to whenever it gets hot -- Iran.
Not sure about what you mean by "the war with Iran has begun". Who's going to war against them? The U.S? Iraq? Both?
If Iran is LOOKING for a pretext that will involve them directly with the US, they have just about managed to poke the hornet's nest just a little too hard.
The name of the game is Muslims killing other Muslims. This would not be of major importance to most of the Western nations, except the conflict does not die down and remain within only their own borders.
There is simply no judicious way to settle this fight. The US is going to eventually have to wade in there like John Wayne as Rooster Cogburn riding out after "Lucky" Ned Pepper in "True Grit".
I'm skeptical of the idea that the Shi'ite government of Iran would purposely destroy a Shi'ite shrine. Also, Sadr is acting like a peacemaker in this situation. If it was his intention to exploit Sunni/Shi'ite conflict, he wouldn't be doing so. There are any number of other players who could have pulled off the bombing of the Golden Dome.
If you don' think our special ops guys aren't in Iran doing their special works you need to read more.