Is time slower on the far galaxies? Since far galaxies move away faster, what would be the speed of their time relative to us? 
If there is a difference:


*

*What determines whose time would be faster? 

*(If I haven't understood it wrong) To resolve the twin paradox, acceleration is required. Is the expansion of the universe creates acceleration? If not how can we explain this difference?


I think following question is the same:
Imagine we have a some kind of machine that can bend the space-time, and create a gravitational pull in front of us. So we gain speed without feeling any acceleration. If one of the twin in the twin paradox would use such a machine to gain speed; what would happen?
I don't have a physics or math background, and I hope my question makes sense.
 A: Marco Ocram's answer is essentially correct; I just wanted to give the formula, and back up with some data:
Time runs slower in distant galaxies by a factor $(1+z)$, where $z$ is the observed redshift of the galaxy. This is a prediction of general relativity, and is observationally verified. That is,
$$
\Delta t_\mathrm{here} = (1+z) \Delta t_\mathrm{there}.
$$
This time dilation applies to all physical processes. The figure below (from Goldhaber et al. 2001) shows how the width supernova lightcurves — i.e. how fast the luminosity increases and decreases — increases as a function of redshift. That is, the more distant the supernova is, the slower it evolves.
The effect is symmetric; an observer in a distant galaxy would see us redshifted, and hence time dilated, by the same factor that we see them redshifted.

The redshift is what is observed. To relate that to a distance, one needs a cosmological model. In the standard, flat "FLRW cosmology", the distance to a galaxy of redshift $z$ is
$$
d = \frac{c}{H_0} \int_0^z \frac{dz'}{\sqrt{
\Omega_\mathrm{m}(1+z)^3 + 
\Omega_\Lambda }},
$$
where $c$ and $H_0$ are the speed of light and Hubble constant, respectively, and $\Omega_\mathrm{m}$ and $\Omega_\Lambda$ are the relative densities of matter and dark energy, respectively.
In the figure below, I plotted the time dilation factor as a function of distance from us. A secondary $x$ axis on top of the plot shows the corresponding lookback time, i.e. how long time ago we see the galaxy; as we approach 13.8 billion light-years — the age of the Universe — time dilation diverges toward infinity. If we could observe all the way back to the Big Bang, processes there would be in extreme slow motion.

A: Time measured in the frame of a galaxy that is moving relative to us would appear to us to be running slowly. The effect is reciprocal, so our time will appear to be running slowly to observers on the galaxy moving with respect to us. For both sets of observers time ticks away at the same rate in their respective reference frames- there is no difference that needs to be explained. The effect of the acceleration is to increase the relative speed at which the two sets of observers are moving, which will have the effecting of increasing the apparent time dilation effect.
A: When you talk about farther galaxies moving faster relative to us, you are presumably talking about the effects of the expansion of the Universe.  When you refer to the twin paradox, you are presumably talking about the special relativistic effects of relative motion.  These are not the same thing.  To see this, note that in the first sense, some of those far-away galaxies are receding faster than the speed of light, so if there were time dilation in the second sense, their clocks would be running backward in our frame.  
Conclusion:  The question is very hard to make sense of.  The recession of the galaxies certainly does not cause time dilation in the "twin paradox" sense that you seem to be thinking of. 
