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Would a dwarf with the travel domain and aspect of the stag(hunter) running, have a movement speed of 35f+ while heavily encumbered in a heavy armor?
Would slow and steady only work on land movement speed or also effect climbing and swimming to since they're related to the movement speed, unless you have a climb/swim speed?
A) Yes, if by aspect of the stag you mean the hunter's class ability Animal Focus(Stag) and not the hunter spell Aspect of the Stag as the spell would get you moving even faster. In your example, your character would have a 35ft base speed that is unaffected by armor and encumbrance.
B) Yes it would apply to climb and swim too. If you don't have a climb speed or swim speed, you follow the rules listed in the climb and swim skills, respectively: Climb at 1/4 base speed, or climb at 1/2 base speed by increasing DC by 5. Swim at 1/2 base speed as a full-round action, or at 1/4 base speed as a move action. Since your base speed is affected by Slow and Steady, it would apply to these as well.