We say pavement in the UK and sidewalk in the USA. An underpass is the same in the UK and the US. Regarding the first question we say "to cross the road" if you mean to pass from one side of the road to the pavement on the opposite side. Using an underpass (if there is one) is safer because you cannot get hit by cars, and yes it is used.

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.