Will gravity pull together two bodies from the other side of an empty universe? Lets say that there are only two bodies in the universe, 65 kg each. Other than that the universe is completely empty, no neutrons, no photons, no dark energy/matter, not even neutrinos (that is to make things less complicated. If the loss of other things leads to something like the universe exploding like a bubble at the speed of light or something, you can change these parameters. I'm mainly concerned about gravity here). Those two bodies are placed apart from each other at the distance of the observable universe. Will they start moving into each other? Will they collide? (Optional question: If so, with how much speed will they collide?) 
 A: I assume a steady-state universe and that the bodies have no velocity relative to each other.
Yes, they will eventually collide.  Gravity has an effect over any distance, including the ~46 billion light-year radius that constitutes the spherical observable universe (the actual size of the universe may be much larger).  Of course, the force will not be very strong over a 100 billion light-year separation, so the bodies would not collide for a very long time.  A rough estimate of the time taken would be on the order of billions of years.
EDIT:  As pointed out in the comments, the above time estimation was wrong by a over a factor of $10^{20}$.  The amount of time taken would be around $10^{38}$ years (100 undecillion years or 100 sextillion years, depending on whether you subscribe to the short scale or long scale).  The equation used to find this number can be found here.
A: Yes, they will collide given the initial conditions.
The speed at collision can be calculated.  We can presume them to begin with virtually zero gravitational potential energy.  We need an assumption of size when they collide.  Let's assume a size of 50cm.  That way when they collide, the centers will be 1m apart.
$$U = -\frac{GMm}{d}$$
When they collide, the gravitational energy will be
$$U = -\frac{(6.67 \times 10^{-11} \frac{Jm}{kg^2})(65kg)^2}{1m}$$
$$U = -2.8 \times 10^{-7} J $$
That's the kinetic energy shared by both when they collide.  Each one has half of that energy since they have equal masses.
$$v = \sqrt{\frac{2KE}{m}}$$
$$v = \sqrt{\frac{2.8 \times 10^{-7} J}{65kg}}$$
$$v = 6.6 \times 10^{-5} \frac ms$$
That's the speed each has relative to their common center of mass.  The time it takes to collide is a more difficult calculation.
A: If the bodies are initially at rest, then the orbit will be a degenerate
ellipsis of finite semi-major axis and eccentricity 1, i.e., a line segment.
The semi-major axis $a$ is half the initial distance. Time to collision is
half the period $T$. This can be directly derived from Kepler's Third Law.
$$ \frac{T^2}{a^3} = \frac{4\pi^2}{GM} $$
$$ T = \sqrt{\frac{4\pi^2a^3}{GM}} $$
If we substitute $a = 46\times 10^9~\mathrm{ly}$ (radius of observable universe), $M = 2\times 65~\mathrm{kg}$ and $G =
6.67\times 10^{-11}~\mathrm{Nm}^2/\mathrm{kg}^2$, we get $T = 6.2\times 10^{44}~\mathrm{s}$. Time to collision is thus $\frac{T}{2} = 3.1\times 10^{44}~\mathrm{s}$, which, according to WolframAlpha, is roughly $7\times 10^{26}$ times the age of the universe.
As noted by others, and developed in BowlOfRed's answer, collision speed may be derived by equating gained potential energy and final kinetic energy.
$$ \frac{Gm^2}{d} = 2\times \frac{mv^2}{2} $$
$$ v = \sqrt{\frac{Gm}{d}} $$
Here $m = 65~\mathrm{kg}$, and $d$ is the final distance, assumed to be much shorter than the initial distance. For, e.g., $d = 1~\mathrm{m}$, we get $v = 7 \times 10^{-5}~\mathrm{m/s} = 70~\mu\mathrm{m/s}$. The relative speed is of course $2v$.
A: if only gravity is taken into account, then when they reach each other after an exceedingly long time they could pass through each other as there is no electromagnetism. I may have missed the point a little and also as a consequence the objects would sort of disintegrate. also would the answer vary depending on the size of the universe?
A: The two bodies will collide at high relatavistic speed, it is conceivable that the actual collision velocity is superluminal compared to each other.
With nothing else to interfere the gravitational attraction will be on-axis. I haven't done the math but 10^36 years sounds high - as the attraction forces increase so will the velocity, and the curves are non-linear. It will take a while to get going though. And that's the cosmological "while".
And we have to ask by whose clock are we measuring the speed and the time? stationary clock in the middle (massless, of course) or by clocks inside each object?
