Are character theme abilities considered Extraordinary (as opposed to Supernatural or Spell-Like Abilities)?
They don't seem to have any notations attached to them like many class abilities do.
Also, what about racial abilities? Do they fall into the same classifications as well, or are they outside that system?
Most theme abilities are not extraordinary. If the ability does not specify a category, then it doesn't have a category.
Abilities without categories are usually
A. Natural abilities that your character has, such as racial vision, abilities gained from your background, and the ability to spend RP to regain Stamina.
B. Class features that allow you to select your abilities, such as talent pools.
C. Spells, they're essentially their own type of ability with its own set of rules