re: Do you accept the notion of the Big Bang as the origin of our universe?

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Before inflation there was no space though (big assumption).

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Well, we don't yet know what existed before the inflation/Big Bang. M-Theory says two branes bumped together to iniate the BB.

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The Big Bang was the inflation of space itself which carried the remaining matter (after the matter/antimatter collisions wiped most of everything out), so since that is space itself that is "blowing up" so to speak it's not traveling through any kind of medium, it would be traveling through true nothingness.

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It's easier to just think of space expanding, rather than trying to account for what it is expanding into. It's impossible to define nothingness without making it something.

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As a result would it be possible for space itself to travel faster than light?

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Guth's Inflation Theory indicates that space can expand much, much faster than the speed of light. In fact, the Hubble Horizon is the edge of the visible Universe. The horizon is the boundary between particles that are moving slower and faster than the speed of light relative to an observer at one given time. Beyond the boundary we'll never see anything, even the light that's emitted.

As space expands faster and faster, eventually everything beyond our local galaxy will be invisible to us. That is unless something weird happens to stop or reverse the expansion.