there is an established command structure and we've already had various types of deployments. We have a rapid response force on alert (it hasn't been called into action yet) and several joint corps.
we did experience a slight shortage (we didn't run out, we just had to shuffle some around) of smart bombs during Libya but we have domestic production that could be ramped up should the need arise.
The EU would also have home field advantage plus almost 50 years of planning for a Russian invasion, we wouldn't need much time to go back into "the commies are coming" mode.
it's not a single cohesive military, but more than enough to repel the Russians even if their military wasn't in shambles (according to their own command).
The French thought they were well prepared before WWII as well. . . After all, Germany was on a leash after WWI, and they had the Maginot line.
Assuming attack by superior ground forces, because Europe does not have anywhere near enough tanks, Europe lacks the air and naval superiority to launch a counter offensive, let alone defend their own airspace or waters. Also Europe does not have a good track record of coming to allies aid quickly enough to alter the outcome in the midst of an attack.
I doubt Russia alone would be the aggressor, however if Russia and China were to decide to return to their old ways of communist hegemony and invade Europe the mainland would fold very quickly. I'm sure they would put up a good fight, but the math would be against them. This may seem far fetched now, however things are not so rosey in Russia that I would discount a possible communist resurgence at some point as people look for an alternative to the political system they have currently.
The time to retool and build your army is not after the battle has begun, you go to war with the army you have.
Being as many of the players in such a scenario have nukes, the question would be whether Europe would launch in the face of such an invasion, or more importantly whether their enemy would think that they would. Europe has been on a moral high horse for quite a while now, I would honestly doubt the credibility of any European nation's threats of MAD.