20 years ago, with a lot more military capability extant, there was a load of talk about the EuroCorps and the Independent European Defense Identity. I was skeptical then, it's a farce to be even talking about it now.
The WEU did put together a maritime flotilla that actually some things done. Two decades ago. Scaling up from that has not quite panned out.
An EU Army, as noted above, first needs a unifying political organization ... which the EU is not.
As to Schengen, I was appalled at what the Europeans were doing when it went down. In some very long discussions with my Spanish, French, and Italian colleagues, I walked the dog on border security and capability, comparing their challenge with the challenges we have on our southern border. Quite frankly, the European problem is a lot more complicated than ours is. Once the internal lines dropped, Schengen as an agreement was one of the surest ways to keep Turkey out of the EU ... so maybe that is why some folks supported it, not sure.
If you don't control your own borders, and if you don't have your own currency, how can you be a sovereign nation?