Can Zombies Climb Stairs?

No but they can pile up and create zombie ramps.  Zombies are the dead risen flesh will rot and decompose so muscle structure is gone.

Yes they can climb stairs and leave blood on them

If theyre super mutated zombies then yes
altho they sometimes can manage smaller steps if not mutated- larger ones are more of a challenge becuase their limbs dont work properly and often fall off (:
Hope thats helpfull

Despite the fact you think they shouldn't be able to, Zombies can climb stairs, especially the 'Runner' or 28 Days Later variety. Zombies are not Daleks, who for years where flummoxed by stairs and will basically try and follow you wherever you go, and cross any obstacle they come across.  

So even though it might not be the first thing you think about while being chased down by Zombies, yes they can climb stairs so make sure you have a plan B ready by the time you get to the top!

Zombies in Left 4 Dead can.