Well, the Western European Union is a part of the EU, and with the ESDP and Permanent Structured Cooperation a European defense pact is slowly taking form which might some day lead to a common Union military.
At the moment it's NATO that has the role of a European coordinated defense. America made it quite clear around the Maastricht Treaty - pertinent was the Bartholomew Telegram - that they are not interested in a break up of NATO or a diminishing of it's role what with the American bases, "special relationships" and interests in East Europe and all that yadda yadda. I am of the opinion that NATO in any form is obsolete, but that it has ceased to be a military organisation as much as it has become an interest organisation, the plans of expansion and "going global" is more than worrisome. The policy of non-European "partner" countries and the NATO mission in Afghanistan shows much of the ambition of it. I don't believe in a organization undercutting the authority of the UN. I fail to see the global benefit. It seems to slowly take the form of a global American toolkit, and I don't see why Europe should hand the US a new pair of gloves. The EU must form a viable alternative to NATO and this is slowly taking form.
Regarding legal jurisdiction with civil and criminal matters, the EU has none. That's all within national control and responsibility. I don't see why the Union should have any jurisdiction there. As the Union is confederate in nature and will be for a forseeable future, it only have powers regarding issues pertaining to EU specific areas put down in the treaties.
As one might guess I am all for the confederacy, and less enthusiastic about a federal state. If that is in the future that might be, but I feel that we must hurry very slowly when it comes to that.