The original EEC was to help maintain peace through economic interdependency. Now the EU is very different with ultimate goal of full integration, i.e. a federal state. The single market is a step, the single currency is another step, EU Parliament, tax rules, now possibly defence matters, etc. Step by step but always in the same direction.
The EU started with the coal and steel initiative, which created a common market for those commodities. It was indeed "peace through trade", but mainly trade.
And the goalpost has since moved very far, I don't see really how preventing countries from sending back illegal immigrants or making same-sex marriage has anything to do with European peace.
The point made is still valid since the EEC was created in 1957 and, before it, the Coal and Steel Community in 1951. Of course it is also unclear what they had in mind with that phrasing, and perhaps it even varied depending on whom you asked: The UK government under Thatcher signed it but I think we can all agree that she wasn't aiming for a federal EU (her famous "No, no, no" speech in 1990 makes it beyond doubt). I think the people at large did not want nation states to disappear within a federal EU, either (and probably still don't).
No. That's false. The EU was supposed to bring peace. The EU "single market" is a mean to achieve peace, not a goal in itself.