Can you take a 5-foot step, double move, withdraw action, or run action while doing the following:
Burrowing
Climbing
Earthgliding
Flying
Swimming
Assume you have the appropriate movement speed (such as climb 30 ft., or fly 60 ft. [good]) for the purposes of this question.
I can't seem to find any rules on the matter and it frequently comes up in our games.
You can take a 5 foot step with those moves as long as those movement types are availible to you.
The fact that the rules say that along with no direct restriction on using the above actions concerning the other movement types means the others are possible. In order for the other movement types to not be allowed to use those they would have to be specifically called out.
Rules text on the 5 foot step.
Take 5-Foot Step
You can move 5 feet in any round when you don't perform any other kind of movement. Taking this 5-foot step never provokes an attack of opportunity. You can't take more than one 5-foot step in a round, and you can't take a 5-foot step in the same round that you move any distance.
You can take a 5-foot step before, during, or after your other actions in the round.
You can only take a 5-foot-step if your movement isn't hampered by difficult terrain or darkness. Any creature with a speed of 5 feet or less can't take a 5-foot step, since moving even 5 feet requires a move action for such a slow creature.
You may not take a 5-foot step using a form of movement for which you do not have a listed speed.
edit:clarification.
Anything else that might restrict the above?
If you somehow enter into a situation that slows your movement for that movement type, but other than that I don't think there should be an issue.
Nice find on the fly DC, Diego Rossi. I did not know that.