Does gravity break time-reversibility on the microlevel?
No, Newtonian gravity is perfectly time-reversal invariant. Suppose I throw a ball up, it falls back down, and my friend catches it. In the time-reversed picture, my friend throws a ball up, it falls back down, and I catch it. Gravity is doing precisely the same thing in both cases: attracting the ball and Earth together.
The reason you might think otherwise is the usual picture of gravitational collapse, like when dust collapsed to form our solar system. This only happens one way because energy is released (e.g. by radiation) as the contraction occurs, increasing the total entropy of the universe. This really is a special feature of gravity: matter under gravity clumps up to increase its entropy, while a typical system like an ideal gas spreads out to increase its entropy. Fundamentally this difference is because gravitationally bound systems have negative heat capacities.
But in any case, it’s the usual macroscopic, thermodynamic arrow of time acting here. There is no microscopic arrow of time in gravity itself.
No, the laws of Newtonian gravity and general relativity are both perfectly invariant under time reversal (although that concept is a bit subtle in the case of GR). Particles don't just "move toward a big cluster and stay there"; the dynamics are generally quite complicated but, in the absence of friction or other dissipative forces, always look exactly the same if you run the film backwards. A quick and dirty way to see this is to note that Newtonian gravity basically boils down to $$m_i \frac{d^2 {\bf x}_i}{dt^2} = -\sum_{j \neq i} \frac{G m_i m_j}{|{\bf r}_i - {\bf r}_j|^3} ({\bf r}_i - {\bf r}_j),$$ and both sides are manifestly invariant under the substitution $t \to -t$.
No. In the case of a cluster of particles, both film directions still describe valid physics. Run the film forward, and one particle moves toward the cluster. Run the film backward, and the particle begins in the cluster with a velocity that causes it to move away from the cluster. That scenario is also perfectly valid physics. The only reason you'd have for deciding that the film was running forward in the first case is because of your assumption of what initial conditions are more likely. That's similar to watching a billiard break backward. The only thing telling you which film direction is forward is your assumption of which initial conditions are more likely. But the physics is reversible.
If you look at just two particles gravitating toward each other, you would not be able to tell which way was forward and which was backward. Release two massive billiard-like particles, and they move toward each other, collide and bounce back. If you saw a film of them getting closer, you wouldn't be able to tell if it was forward (attracting each other, starting from rest) or backward (recoiling after the collision).