Did space exist prior to the big bang? Does not matter create its space? Can space exist without any matter of any kind?
Very good questions. Some of them have clear-cut answers, but others are very deep and it's difficult to give a satisfactory answer.
Did space exist prior to the big bang?
Most scientists would answer “No” to this question. The Big Bang marks the beginning of space and time as we know them, so there was no space before the Big Bang. I say “most scientists” rather than “all scientists” because the Big Bang itself is not completely understood, so it's possible that our understanding of this event will improve over time.
Does not matter create its space?
I'm not quite clear on what you mean here, but if you're asking “Can matter exist without space?” then I suppose that the answer is “No, at least not as we know it.”
Can space exist without any matter of any kind?
This is an extremely deep question, and people have been arguing about it for ~100 years. Albert Einstein's thinking about this question led him to formulate General Relativity! Part of the reason that such a seemingly simple question is so difficult to answer is that there's no way to do an experiment to find out the answer. In order to do such an experiment, we would need to create a whole new universe that contained space but no matter, just to see if it's possible. Unfortunately it's rather difficult to create a whole new universe!
If you're interested in (sometimes heavy) further reading, look up “Mach's Principle” (http://en.wikipedia.org/wiki/Mach's_principle). In discussions of Mach's Principle, people usually use the phrase “inertial reference frame” to mean basically the same thing that you mean by “space” here.
The upshot is that it's completely unclear if one can have a universe with space but no matter, or if one needs matter in order to basically serve as reference points and give meaning to distances.