I recall reading that you can Run with any movement speed type that you have.
As in, even if you can climb, you cannot Run while climbing unless you actually have a Climb movement speed.
As to movement downward, if you are going diagonally downward you do not double you whole movement, you pretty much only get a bonus 5ft for every 5ft of lost elevation.
Aye, took me a while to roll the checks. Got surprise stuff to deal with between posts. >_<