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Often while playing Pathfinder, my group will run into problems where rules for 3-dimensional play simply don't exist, so far as we can find. Arguments will come up over things like carrying-capacity, movement speed, area-effects, and a variety of other things.
Simply put, are there rules for 3D play for Pathfinder, and if so where can I find them?
The game does have 3D rules; they’re long and annoying, and mostly given in William’s answer. I’m not at all clear what more you want.
Personally, I despise these 3D rules: they’re difficult and time-consuming to use, because there’s no good way to indicate 3D area-effects, calculating distances moved involves Pythagoras and marking the paths used is impossible, and so on. They represent a bizarre and considerable increase in the amount of simulation that the system offers: at the same time that all of the maneuvering and positioning and lunging of combat is abstracted to a five-foot square, you have flight speeds that have turning radii, minimum forward movements, different speeds for up and down, and the aforementioned necessity for using the Pythagorean forumula to figure out the diagonals. These are details that I do not think the game needs in the slightest; they are way more detailed than most other aspects of the game. I want an epic fight against a dragon, not an incredible dragon-flight simulator.
I dislike these rules so much, in fact, that I have written houserules to eliminate most of them altogether. There are still nods to the differences in speeds for going up and down, and I do use the minimum forward speeds and turning radii since otherwise the maneuverability ratings wouldn’t mean anything. But flying or not is just a “status condition” that changes how you interact with, say, difficult terrain, and whether or not people can melee you or vice versa. It eliminates all need to figure out 3D distances or shapes, and is therefore, in my opinion, massively superior.
Just an alternative for your consideration.
I'm not a Pathfinder player myself, but this place seems to cover all you've asked and it has a search feature.
I'll try and answer the few specific questions in your question. Flight and 3D movement isn't covered by one general ruleset, and it's more specific to the creatures or means of flight.
Carrying Capacity does not mention specific rules for altered capacity when flying. This leads me to think it's covered in the means of flight rather than a generel rule governing it. Which makes sense as there are several means of flight, both magically, super-natural and extraordinary. Fly(spell) and Overland Flight(spell) shows you can carry up to heavy load, though at a reduced speed. The Fly(skill) which dragons and other natural flying creatures use, doesn't mention anything different in regards to carrying capacity. In the comments below you mention that it doesn't make sense that four-legged winged creatures can't carry more than two-legged winged creatures. I think it's assumed that creatures with four-legs also have stronger wings. If they need four-legs to support them walking, they'll need the stronger wings to carry them flying. For magic, it simply scales to match what a similar winged creature would have.
Movement when moving in 3D, that is flight, is covered by the Fly(skill). It covers maneuvers and the skill checks needed.
Aiming a Spell covers the shapes of area of effect spells. Those extend similarily in 3D. A spread will move out and around obstacles along the max distance it spreads. A burst will hit everything within the the range. This can be hard to visualize in your mind and sometimes you simply have to pull out the calculator and check the distance using Pythagoras.
