Does Shepard's background (Origin and Reputation) matter?
Solution 1:
I played a Spacer War Hero in my Mass Effect 3 playthough, so while I can't speak for the other origins, the Spacer Pre-Service History (mentioning Shep's mother, who's captain of another ship in the Alliance Navy) came up a lot less than the Background (Hero of Elysium against the Skyllian Blitz) did.
For the demo in particular, it's practically irrelevant; the only reason it's in there at all is because the demo features the tutorial level, and it was probably more trouble than it was worth to remove it.
Neither has a huge change on the story - in many ways, it had the most effect(ha!) in Mass Effect 1, simply because each Pre-Service History had an associated assignment, while later games did not.