Does object from which the universe is created still exist in its original form? If time is passing more slowly near objects with bigger gravitational mass, could it be that the time at the object from which the universe is created has not started yet (or very little of time has passed on it), and that such object still exists?
If Earth core is 2,5 years younger then its surface, how much younger is place from which the universe had started? 
 A: According to the standard Big Bang theory, there was no "object from which the universe is created" distinct from the whole universe itself. Similarly, the "place from which the universe had started" was the whole of space, although space has stretched itself quite a bit since then. 
Although space is expanding, it's most likely infinite in extent, and if so, it's been infinite since it came into existence at the start of the Big Bang. It's not easy to conceptualize an infinite thing expanding, it might help to think of it in terms of its mean density reducing over time.
But apart from those rather important issues, your logic is sound: the age of the universe when measured from the frame of reference of an ancient very massive body would be less than the age estimate of an observer who's far from any sources of gravity.
However, to the best of our knowledge, there were no large bodies formed in the very early era following the Big Bang. In fact, it was so hot that not even neutral atoms were stable until the universe was around 380,000 years old, so things bigger than atoms were unlikely to have formed before then. 
There's some recent evidence that stars were beginning to form when the universe was around 180 million years old. Those stars were very large, and so they burned quickly, and none of them remain in the present era. Some of them may have turned into black holes at the end of their lives, but some would have just blown themselves apart in a pair-instability supernova, depending on size. Any black holes formed back then still exist, either as individuals, or merged with other black holes (and whatever else got in the way) to form larger black holes. 
I'm sure you've heard that most galaxies have very massive black holes at their cores. We still don't have a good theory to explain why they're so big: there doesn't seem to have been enough time for them to gather that much mass. Presumably their formation started relatively early, but we still don't know exactly how early.
Time does get rather dilated in the vicinity of a black hole, but the situation is quite complicated, so I won't go into it here. Besides, there are already many questions on this site about that topic, we probably don't need another one. :)
A: You appear to hold a very common misunderstanding that is often reinforced by the term "big bang". The term suggests that something exploded and the universe that you see today - the stars and galaxies and such - are the bits that went flying off it.
This is not the case. It is not the stuff inside the universe that exploded, it is, in effect, the outer surface of the universe smoothly expanding so there is more and more room inside. As the room grows larger, the temperature drops (quick: what is the definition of temperature?). Eventually it is cool enough that solids can form, and presto, you have the universe you see today.
Think about a balloon that you put in a vacuum chamber. Originally the air inside the balloon is the same temperature as the air around you. Now as you remove the air from the chamber, the balloon expands, and the air molecules inside spread out and cool off (again, definition of temperature). This is what we see around us in the universe, it is cooling off and spreading out. We're not 100% sure why it is expanding, but it's clear it is.
If you're a single air molecule in the balloon, you did not "come from" any original object. You were always there. Your relative position within the balloon didn't change, but as the balloon itself expanded, the space between you and everyone else got larger. This is very different than, say, you were originally part of a drop of water that was explosively heated.
So you're asking about "that thing", but there was no thing. The universe is not a thing, its the line we draw around the collection of things.
