Fate of largest scale structures? In $\Lambda\mathrm{CDM}$, structures form "bottom up" with larger structures forming later. Structures are generally speaking supported by the velocity dispersion of their constituent objects (e.g. elliptical galaxies are supported by velocity dispersion of stars while galaxy clusters are supported by velocity dispersion of galaxies)$^1$. More massive virialized structures require higher velocity dispersions to support them. What happens when the velocity dispersion required to support a structure becomes relativistic and eventually exceeds $c$? Does the structure simply fail to collapse? Collapse to a black hole? Something else?
It occurs to me that the $\Lambda$-driven exponential expansion that is currently thought to be getting under way in our universe might be rapid enough to cut off structure collapse at some scale, avoiding the scenario I described above. For the purposes of this question, let's assume for convenience a model where the Universe continues to expand with $\lim_{t\rightarrow\infty}\dot{a}(t)=0$.
$^1$ With the notable exception of systems where dissipation is important, allowing the formation of a rotationally supported disk.
 A: They will form either separate universes or black holes.
This is a really interesting (and old) question. Let me give a little more background before I explain my reasoning for the answer. As the question specifies, we consider a universe without dark energy.
Background
Collapsed, velocity dispersion-supported systems can grow continuously as they accrete new material. In general, they are $\Delta_\mathrm{vir}\sim 200$ times denser than the cosmological average (the "virial overdensity"). So a spherical system with radius $R$ has mass
$$
M
= \frac{4\pi}{3}\Delta_\mathrm{vir}\bar\rho R^3,
$$
where $\bar\rho=3H^2/(8\pi G)$ is the cosmological average matter density, $H$ is the Hubble rate, and $G$ is the gravitational constant. Such a system could in principle grow without limit. But the gravitational potential depth at the edge of the system is $|\Phi|\sim GM/R$, and the velocity dispersion of the system is $\sigma\sim\sqrt{GM/R}$. Thus, both special and general relativistic effects become important when
$$\frac{GM}{R} 
\equiv \frac{\Delta_\mathrm{vir}}{2} H^2 R^2
\sim c^2.
$$
This corresponds to $R\sim 0.1\, c/H$, that is, the system's radius is a tenth of the Hubble length. So we arrive at the question. What happens when our system grows to this size?
Separate universes and black holes
A similar idea has been considered in a more idealized context. If the density of the universe were to exceed the critical density, then it would have finite size and loop around on itself like a sphere. This universe would be said to be closed. But what if the density of our universe were exactly the critical density (as it appears to be), but inside our universe, there is an overdense region of space whose size exceeds the size of the closed universe corresponding to that overdensity? It would seem that this region of space should then be a separate universe, completely causally disconnected from our own universe.
The formation of such a separate universe was explored in detail by Kopp, Hofmann, & Weller (2011). There are some very neat diagrams in that paper, which I recommend at least glancing at. For example, here's an embedding diagram showing our universe (top) and the separate universe (bottom) shortly before the latter "pinches off".

However, the separate universe eventually collapses, since it exceeds the critical density. One of the conclusions of Kopp, Hofmann, & Weller (2011) is that due to this collapse, the conformal structure of the separate universe-forming spacetime is identical to that of a black hole-forming spacetime. The disconnected regions of space only arose because of the choice of time slicing (which was chosen to be synchronous, i.e., comoving observers on the same slice say the same amount of time has elapsed). In an alternative time slicing, they would not arise. So in some sense, forming a separate universe is not fundamentally different from forming a black hole.
This analysis suggests that our structure, which is $\sim 200$ times overdense and growing in size, should eventually become a separate universe, which is in some sense equivalent to becoming a black hole.
The velocity dispersion
But the separate universe analysis didn't consider a velocity dispersion. Can a velocity dispersion support a universe against collapse? If so, then perhaps a static separate universe could arise, which would be fundamentally different from a black hole. I do not know the answer to this, but I can give a few considerations.
On the one hand, cosmic structures grow from the inside out, with newly accreted material primarily remaining at large radii and not affecting the interior structure. As our object accretes material beyond the threshold to form a separate universe, the new material remains at large distances, and there is no reason to expect that it should influence the interior at all. In particular, it should not cause the interior to collapse.
But that's the Newtonian picture. Relativistic considerations might change it, whether special (e.g. the speed of light limit) or general (e.g. the briefly infinite blueshift of the continued accretion from the outside universe). Also, in black holes, there is a violent instability that arises in rotating or charged black holes from the interaction between infalling and outflowing material at the inner horizon (Poisson & Israel 1989). I wonder if the velocity dispersion inside the separate universe might induce a similar effect.
In the end, I think it is fair to say that the structure will form a separate universe. Whether that universe promptly collapses (so we could just say we formed a black hole) or remains stable is unclear.
A: This question cannot be answered objectively:
(1) Am I right that you are assuming a "static universe" with lambda to zero? That also means that you neglect Hubble constant (accelerated expansion) and ca. 73% dark energy of the LambdaCDM model. Your assumption contradicts reality.
http://en.wikipedia.org/wiki/Lambda-CDM_model
(2) Cold Dark matter is responsible for structures in the Universe, but nobody knows so far what Dark matter really is. Can it move faster than light? Nobody knows. 
https://www.youtube.com/watch?v=lNNOaYwiu_U
(3) No spinning structures with velocity bigger than c has been observed. And in reality (without your assumption) our fate will be: our galaxy will merge with Andromeda galaxy and all other galaxies will be out of the event horizon. (So to observe your case becomes more and more unlikely).
http://en.wikipedia.org/wiki/Velocity_dispersion
https://www.youtube.com/watch?v=SMSQu7C2gFA
