Do Iraqi Soldiers have the will to fight for their country? Do they need more training?

This is important stuff, so please forgive the diatribe.
What gives US soldiers the will to fight for their country? Lots of things, including the internal belief in the goodness and rightness of America and the constitution and the close relationships and team camaraderie with their crew or squad. How we grow up in the US, believing everyone is equal, helps us make those close linkages within our squads and crews, regardless of what tribe or religious sect or skin color the squad members might be. And even in America, there are dysfunctional units and racist or biased members that prevent the close cohesion needed, so it can be hard to get over those "tribes", even in the US. The Iraqi people didn't grow up like us, and I fear no amount of training will overcome the fundamental internal tribal/ethnic/sect identity they have of themselves. Ask them who they are, and they will more likely identify as "Sunni" than "Iraqi". This is why American fighting forces are so strong...we identify as American above all and can form close bonds with our brothers and sisters in arms based on that identity.

No, they don't.

They don't identify themselves as Iraqis, and until THEY do, no amount of equipment or training will get them over that cultural hill.

Someone else wrote about leadership - that too is missing. Nouri al-Maliki and his fellow govt members had a chance to start building on IRAQ. Instead of demonstrating courage and fortitude, following an example laid down by Nelson Mandela, they chose the time-honored tradition of cultural tribal/religion rules all approach.

Until the facts on the ground bear out something other than that a ragtag band of pirates routs them from city after city, it is difficult to assess any other way than they lack will.
Training would help.
Better leadership would help.
Better support would help.
But without those things, you would think that they would at least be even in a set-piece fight, with raw numbers heavily favoring the IA.
Clearly, the terror and psychological tactics of IS are causing the IA units to break and retreat.

They don't need more training. Collectively, they need to grow a spine. In Iraqis' defense - they don't think of themselves as Iraqis first, like Americans do. Their first loyalty is to their tribe or clan, then town. It's reflected in the name - Sadaam Hussein blah blah al-Tikriti, for example. Having trained some of these people, I now they'll fight like hell for their tribe, clan or town, but the idea of a country is foreign to them and one not easily understood.

They have the will to fight-- that will is just an untapped resource. The Shia'a and Kurdish militias, and even ISIL itself, shows that the people there can fight if they feel they have a cause worth fighting for. It may not be a cause WE believe in or like, but it motivates them.

The problem for the Iraqi people is that they view their government as corrupt, inefficient, and unresponsive to their needs; they also don't have a sense of allegiance to the "Nation of Iraq" as a concept. The borders were made by Europeans (British & French, mostly) after the fall of the Ottoman Empire, so "Iraq" reflects British intentions for the region, not local interests.

Unfortunately, when Americans show up and give them equipment and training, even if we have good intentions it gives the appearance that the local government is just a "puppet of the imperialists" or some such thing, which is hard to rationalize for them. Especially since we insist on rebuilding Iraq the way it was.

Ideally we'd very quietly arm the sides we prefer over ISIL, and let them fight it out on their own with little obvious interference on our part. That may mean holding our noses and working with Shi'a groups, but they're better than ISIL. And who knows, maybe we can use that as a bargaining chip with Iran in some way.

If what General Petraeus says is true then they need more training. I would also add stronger leadership within the ranks to lead the fight.

An Iraqi LTC told a US CPT that I know once to give up on the Saddams Army vets and just start over with a new generation

I think it really varies wildly from province to province and town to town. The Kurds will stomp you hard your grand kids will feel it and they give no quarter. Um Kasar? Couldn't give a shit. Baghdad proper? You can have Bagdhad when you pry it from their cold dead hands. Taji? Eh, maybe we'll just move somewhere else...

The National Identity of Iraq is still quite fractured, and not from American influence. They only were united by fear under Saddam.

When we look at the history of our country, we often forget that the Revolutionary war was the START of our statehood, not the cementing of it. We had more conflicts - some war, some not - and more alliances that brought us to wear we are. Hell, the first country to recognize America as a sovereign land was Morocco.

You can train and equip a person all you want. If they do not have the intestinal fortitude to fight, it isn't going to happen.