There is no evidence it did.
There is the "zero energy universe" hypothesis: https://en.wikipedia.org/wiki/Zero-energy_universe
There is not evidence that space-time is quantised (despite experiments to detect it).
No.
A better analogy is that space is falling towards massive objects: https://jila.colorado.edu/~ajsh/insidebh/waterfall.html
(But it is just an analogy. Space is not "stuff".
Not unless you have math, quantitative predictions and objective tests of those predictions.
If there was nothing, where did these fluctuations happen?
There is no evidence that spacetime has mass, and no evidence of anything with negative mass.