Is there already room for the expansion of our universe since Big Bang。But the prerequisite is there has to be space in existence in the first place to allow such expansion。?

No, that is exactly the mistake so many of us make: it is not an expansion into something. If it was, then we should ask - as you do - what is that being into?

We tend to see the universe as we do for an object on earth: We see it from the outside, relative to another object and all of it simultaneously.

You have seen illustrations of the universe as a cone where the big bang is the apex. You have seen illustrations of the universe as an inflating sphere where galaxies are spots on its surface. I will now suggest a third illustration:

That of the observer being at the center of a sphere the surface of which is the big bang. Strange? Well, think of it: we see the sun as it was 8 minutes ago. The closest star as it was 4.2 years ago and, distant galaxies, as they were billion of years ago. The farther we can see is ... the big bang, and we can only observe the universe as being its center, just like a fish swimming in the ocean of a planet without continents is always in the center of the ocean.

Yet none of those three illustrations is correct. We simply cannot imagine the universe as our brain is made for: calculate distance and time, needed for the hunter-gatherer, we still have the genes of, to catch its prey.

The universe has no center, no boundary and no outside. We cannot "imagine" it, we have to accept the scientific facts otherwise it doesn't make sense: If the universe was expanding into something, what was the limit of that "something" and so on! The same can be said about time: if time had to start with a "creator," who created the creator and if he can be eternal, why not the universe?

The universe isn't actually expanding at all. The idea that it does comes from misinterpreting galactic redshifts as a doppler effect when they are actually a scattering effect. The galaxies are not generally receding from each other — their light simply loses energy through it's interaction with the molecular hydrogen that fills intergalactic space (see source).

If space expanded, as well as stretching out the energy of light radially causing distant galaxies to appear redshifted, it would stretch it out transversely causing a reduction in their surface brightnesses. This is not what we observe:
http://www.sci-news.com/astronomy/science-universe-not-expanding-01940.html

Expanding space would also stretch out the light curves of quasars (their oscillation in luminosity). No sign of this either:
http://phys.org/news190027752.html

This documentary may enlighten you:
https://www.youtube.com/watch?v=IFFl9S39CTM
There never was a Big Bang.

The big bang theory does not make any claims about the existence of space. The theory only describes how space has been expanding and cooling since the Planck epoch, which was at least a few fractions of a second after the expansion had to already be underway, until now. At no point does anyone claim this was the "beginning" or the "creation" of space. Space (the universe) very well may have always existed, and even always been of infinite size. It just hadn't always been expanding. At least not here in the region we are in.

No, there can't be space in existence in the first place to expand into. If there was, it would already be part of the universe.

The universe is expanding into nothingness, and creating its own space, as it expands.
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