Of course there are huge problems with this -- what happens if whomever makes your food goes tits up, involuntarily (natural disasters, coups, ...) or voluntarily (maliciously attacking)? Who guarantees the system? How? Etc... --, if there weren't I wouldn't call it utopia. But I think raising the costs of war beyond the unreasonable is a net gain.
Note: I'm fully aware that this is an extremist and probably next to impossible position, no one has to remind me that. I just contest the fact that the objection "but then we could not go around and kill people" isn't a particularly good counter-argument for no trade barriers.
There are no tariffs on food, steel or similar between the member states. All states subsidise food production, but not steel.
If you mean you'd like to see the US and those countries both drop tariffs and protections, I can see you argument - but that's the opposite of the direction things are going today so I don't see the relevance. Even if it were not, it's hard to argue against a countries legitimate security concerns in some areas, although clearly that argument can be abused (again, cf current rhetoric).
The EU is a really good example here of how a single currency puts everyone on the same economic accounting basis and so eliminates the ability of a country to 'fudge' the numbers through either internal subsidies or currency manipulation.
The reason you need a single currency is that if you're going to sell barrels of wheat from country X to country Y, and country Y buys televisions from country X. How many units of currency is a television worth relative to a barrel of wheat? If those values are tied to the same currency, then the economic incentive to grow wheat or build televisions will be on the same footing.
A tariff is a way to adjust this economic cost externally which takes it out of the market's hands and put it into a political realm and one step removed from the actual economy that should be adjusting.
In a single-currency market, it is the responsibility of strong players to shore up the economies of weak players, or risk popular discontent in the weaker economy and the democratic dissolution of the common currency. German politicians have paid lip service to this but have not really put German resources to use in building up the PIGS economies because such a policy is unpopular with the German voting public, which wants to see those resources stay in Germany and to the nearsighted benefit of German taxpayers. This is the real root of the Euro crisis. There were always going to be weaker players in the EU, and it doesn't matter if it's Greece, it could very easily have been some other country. The question is what do stronger players do.
The US doesn't suffer from this issue because, to use an oversimplified example, California voters aren't discontent with FEMA funds used in hurricane disaster recovery areas. Californians are Americans before they're Californians, by and large.
This is true, however there were quite strict criteria (the Maastricht criteria) put in place to try and prevent this from being too bigger problem. In hindsight, those criteria weren't sufficient, in one way or another.
I believe it's correct to say those weaker economies benefited from adopting the Euro the short term, and that they wanted to join the Euro.
As a twist, the Euro was somewhat unpopular in Germany when it was introduced, because it made things more expensive. So there is a feeling that Germany "paid" the price of adopting the Euro - whether this is true or not is irrelevant for the sake of policy-making.
In general, freely floating currencies fix all problems you mention.
Tariffs and currency manipulation are a sovereign response to prevent efficient markets that would render local labor noncompetitive.
And as I mentioned earlier, a single currency makes such manipulations ineffective and you get Greece. They can't bring their productivity up to the level of Germany or France and their economy suffers for it because all of the capital goes elsewhere.