I'm not sure how you could use a pavement to cross a road. The pavement usually runs along the side of a road. Or are you referring to walking on the road itself?
Here in New Zealand we use pavement to mean the same thing as footpath in British English. I think the Americans use 'Sidewalk' for that.
Yes, it would be natural to say take the underpass to get to the other side of the road. Here, we don't have a rail subway system, so we would also say 'use the subway' to mean an underpass to get to the other side.