Does the unobservable universe exist in the present? Is our observation about universe limited due to the incapacity of telescopes to look further or is it just because  we look too further into the future, so the "unobservable" universe hasn't been born yet?
 A: Our telescopes don't look into the future, but rather the past. In other words the light which we observe has been on a long journey to Earth for a long time. Some stars are closer so their light is relatively younger. 
The closest stars are Alpha Centauri A and Alpha Centauri B, which form a binary pair. They are an average of 4.3 light-years from Earth. So the light from them has traveled 4.3 years before getting to the earth. 
MACS0647-JD is a candidate for the farthest galaxy. Based on the photometric redshift estimate, of about z = 10.7 - 11, the equivalent distance is 13.3 billion light-years. So the current structure of that galaxy is very different today than what we are observing now. We are only seeing what it looked like 13.3 billion years ago. 
Even our sun is a long distance relative to earth distances. It takes about 8 minutes and 20 seconds on the average for the light from the Sun to get to the Earth. 
A: Ok, this is a funny question.

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*If I wanted to be picky I would ask: According to who's present, as this notion depends on the observer (due to relativity). Let's assume for now that you are talking about your present to get this out of the way.


*The Observable Universe has a boundary as there are regions that are so far that the light did not have time to reach us from there since the Universe was born. This means that we have absolutely no information about the present or even the past of those places. (There are many ideas what there could actually be, but there is no widely accepted theory to my knowledge.)
I tell you better. Even on a sunny afternoon you cannot be sure if the Sun still exists as it takes light 8-ish minutes to reach us from the surface of the sun as MaxW pointed out. This means that if our star suddenly disappeared we would have zero clue for this amount of time. This would be a very pathological case though.


*At the same time this boundary is continuously expanding and in theory we should be able to see more and more every time we look up on the sky. The only problem is that the new things are gradually revealed. You have to watch this movie from the beginning, so what we can see there are the remains of the birth of the universe, which is mostly the cosmic background radiation (CMBR), and is not visible.
(Take this with a pinch of salt, the early universe was actually not transparent, so you actually see the distorted image of a later stage in CMBR.)
(Also I am not an astrophysicist. Consider yourself warned.)
A: First you have to decide what "exists in the present" means.
This seems like a reasonable definition:  Let $E$ be the event "here and now", choose a frame $E$ in which you are stationary, exponentiate that frame to get a (maximal) coordinate patch $S$ on spacetime, and let $P$ ("the present") be the (three-dimensional) locus in $S$ of $t=0$. Then to say that "the entire universe exists in the present" is to say that on any timelike geodesic, there is a point $x$ and a lightlike geodesic passing through both $x$ and some point in $P$.   In other words, any object that thinks of itself as "stationary" spends a moment in the present.
Whether or not this is true depends on the topology and geometry of spacetime.  It's obviously true in Minkowski space, or more generally in spacetimes of the form $S\times T$ (where you have a global time coordinate).  It's also easy to write down examples where it's false, e.g. any spacetime in which there is a timelike geodesic disjoint from the maximal coordinate patch $S$. 
A: This is a very interesting question, because we have to think about both what constitutes the "observable" universe and moreover what constitutes the "present", which actually bumps up deeply against relativity theory.
In particular, in at least special relativity theory, one of the things that is known is that you can only say unambiguously that "this event is later than that one" if the spatial location of the two events are not so far apart that a light beam emitted from the early event toward the later one will be unable to reach the later one in time to witness its occurrence. For events that are too far apart, it is possible to have an observer who can conduct measurements showing that the "later" event happened "before" the "earlier" one. Such an observer is distinguished from the one assigning them the first orientation, of course, by hir state of motion. In addition to that, there is a third observer who will measure both events to occur simultaneously, and thus effectively when two events are such "too far apart" (the technical term is "space-like separated"), the chronological ordering is either undefined or in a sense you could say all of them are in the "present", forming a "thick" present (effectively as thick in years as the thing is far from you in light-years) or you could say the concept of "present" is not a well-defined one.
It gets even worse in general relativity, where we warp the spacetime fabric and then the kind of simultaneity measurements we are talking about there are problematized even further, to the point that you cannot really assign even an observer-dependent simultaneity present across an arbitrary curved spacetime. (You can still do it approximately over scales where the curvature is not too large and so it still "looks mostly like" the spacetime of special relativity, but not more than that. Fortunately for us, the Universe appears globally flat, however there is likely problematization due to its expansion which effectively renders it curved in time.)
The events which a given observer can unambiguously say are in hir past and future form a sort of "cone" in 4-dimensional space, called a light cone. The outside of this cone is the "thick present", and consists of the events space-like separated from hir. The edges of this cone represent respectively an imaginary expanding sphere of photons, which was emitted both pastward and futureward (both physically meaningful due to time reversal symmetry) at hir present moment-place (i.e. both hir current time on hir watch and hir exact position in space.).
Now if we move to the context of the Universe, naively the observable universe is all that within the past light cone of us on Earth - or even better, on its surface (since that represents all paths of photons journeying from other spatial positions to us). The unobservable universe thus lies outside that cone - either in our future cone, or in the "ill-defined temporality" region that I called above as a "thick present". If you use that then the answer is a sort of yes: it exists in the thick present, but also in the future. As we "move" forward into time, our past light cone sections through this region and thus things become observable, however due to the Universe's expansion there are regions that will never be.
