Run

Can a character run with other Speed, such as Swim, Fly or Burrow?

Can a character run with other Speed, such as Swim, Fly or Burrow?

Swim - yes, per the Swim skill

Climb - no, per the Climb skill
Burrow/Earth Glide - no, unless you have some crazy senses. Per the Run action, you must be able to see where you are going to run.
Fly - Maybe. The fly spell specifically says you can charge but cannot run while flying under the spell's effect. However the Fly skill is silent on the case of creatures with a natural flight speed.