yes you can jump. hold down the button to run "B" and whilst running quickly press B again. it is not an exact science but it does work.

Most broken stairways (might be all, so far I've found none) you can't jump up. There will be another way around if there is something up there.

If you mean in the sense you'd think in most games.. press 'A' to jump.. No, you cant.
As for the closest thing to jumping, what Crayonmuffin said. :)

To jump you have to first run by holding B button down then tap the B button again when your running.

The distance you jump will be determined by how fast you are able to run. And how fast you are able to run is determined by the weight of all your equipment.

If the weight of your equipment is less then 25% your movement speed will not reduced.
25% to less then 50% - 25% speed reduction.
50% to less then 75% - 50% speed reduction.
75% to less then 100% - 75% speed reduction.
100% not sure never exceeded my equip load.

To put it simply, no, but you dont need to anyway, Cause you are going to be rolling like a king ;)

If it;s the broken staircase in the Undead Asylumn, you get to return later w/a key that let's u access the item (it's a ring) late. Hope this helps