I think it's very likely that military operations will become more and more coordinated. After all, they've already pooled their nuclear programme, and although the Eurosyn does serve a political/economic function it is predominately about Cold War diplomacy. At this point there's basically no need for the ES nations to maintain individual forces, except I suppose the 'flagship' projects and small stuff for local work, as you say. Really the only possible zones of contention are going to be the GIUK–North Sea–Baltic area, where Germany gives the US a possible in, and maybe I guess pushing into the Atlantic beyond Sole/FitzRoy? Idk, I suppose ti would rely on either Eire or Portugal going rogue…
Given the geographical situation, Eurosyn can easily power project through the Med and Atlantic, and have the Soviets in a death grip navally except in the far Pacific. Thus, going to be lots of use and utility in having a unfied fleet building program.
Given all that, the power and threats to Eurosyn are land based. So air forces and land armies are going to be increasingly linked together. I wouldn't be surprised if the continental members just joined all their armies together. Might as well, given they're all going to be along the german and Soviet borders anyway.
Other joint projects are the nuke stuff as already covered, and probably cruise missles as well. As I said, I think that only things that the big countries won't share with the rest/each other are the big three: carriers, subs and monetary policy.