Can speed of light vary from one to another region in universe? I think speed of light and property of spacetime and gravity are interrelated. So is it possible that the speed of light in the past universe was more/less compared to now?, Does the property of spacetime varies from place to place?
 A: The answer depends exactly what you mean by "speed of light" and relative to what. 
For every observer, the speed of light in their momentarily co-moving inertial frame is always exactly $c$, in general relativity. There is no variation from that standpoint. This means if you time a laser pulse between to nearby points in a laboratory that is comoving with you, you will always measure $c$, no matter when and where you are in the Universe.
However, when you measure light's time of flight between distant bodies, then the speed of light can indeed vary as determined by your clock and known distance between those bodies and you can find situations where this measurement is any positive value. An instructive example is the Schwarzschild metric for a nonspinning, noncharged black hole:
$$c^2 \,{d \tau}^{2} = \left(1 - \frac{r_\mathrm{s}}{r} \right) c^2 \,dt^2 - \left(1-\frac{r_\mathrm{s}}{r}\right)^{-1} \,dr^2 - r^2 \left(d\theta^2 + \sin^2\theta \, d\varphi^2\right)$$
and whose $t$ co-ordinate represents the measurement made by a clock infinitely distant from the black hole, where the spacetime is flat. $\mathrm{d} \tau$ is the proper time interval. As an observer $A$ freefalls through the horizon, sending light pulses to a distant obsever $B$, the latter $B$ (whose clock measures $\mathrm{d} t$) infers that the speed of those pulses slows down more and more as $A$ approaches the horizon. That is, suppose $B$ wants to know what the duration, as measured by his or her clock, of a unit length, lightlight interval between two distant neighboring points is. So $B$ puts $\mathrm{d}\tau=0$ because, being a good GR student, they know that lightlike intervals have zero length. $B$'s clock's measurement of the delay $\mathrm{d}t$ for a radially ($\mathrm{d}\theta = \mathrm{d}\phi=0$) directed light pulse in travelling a unit length, radial distance ($\mathrm{d}r=1$) is then given by:
$$c^2 \,{d \tau}^{2} = 0 = \left(1 - \frac{r_\mathrm{s}}{r} \right) c^2 \,dt^2 - \left(1-\frac{r_\mathrm{s}}{r}\right)^{-1} \,1^2$$
which means that $B$ sees the light pulses travelling at speed $c\,(1-\frac{r_s}{r})$. That is, locally $B$ measures lightspeed to be $c$ ($r\to\infty$) whereas light slows down as $r$ becomes nearer to the Schwarzschild radius, at last stopping althogether as $r\to r_s$. A related fact is that, from $B$'s standpoint, a freefalling black hole base jumper never quite reaches the horizon, whereas the base jumper coasts through the horizon in finite time as though nothing special has happenned. You can see this fact manifested as the divergence of $\mathrm{d}\tau / \mathrm{d} t$, which becomes infinite as $r$ approaches the Schwarzschild radius $r_s$.
So when we look at distant things in the universe, i.e. things from former times, we may indeed observe a varying lightspeed in gravitationally affected situations.
But you may be asking this question owing to reports, in the popular press, of theories where $c$ had a different value from what it has now. These, unfortunately, are beyond my ability to comment on. You need to ask, "relative to what"? If you're talking about the delay of light pulses measured by any past observer in a momentarily comoving inertial frame, then this value can only vary from the universal $c$ if either:


*

*Light in the past was assumed to have rest mass, if standard general relativity is assumed to hold; or

*General Relativity is not accurate, and some beyond-classical-GTR-theory is being contemplated.


All kinds of variations on GR are studied in an effort to explain the Horizon Problem through introducing a Variable Speed of Light in various ways. Whilst these theories are possible, and are postulated as an alternative explanation to cosmic inflation to explain the Universe's homogeneity, I do not believe we have firm experimental evidence for any of them yet.
