chargeflightmovementpathfinder-1e
Flying is a common ability in Pathfinder; so is using special attack options like charging. What's not clear is when they're used together.
Are you able to charge while flying? Can you do so while burrowing, swimming, climbing (provided you have the correct movement for it), or other special movement types which have a specific speed that don't require extra movement?
If you lack a swimming/climbing speed, you use your current walking speed to swim/climb but must spend extra movement. The swimming/climbing rules don't shut off increases to your walking speed.
When you see the speed on your character sheet or in a monster stat block that is walking speed
Every character and monster has a speed, which is the distance in feet that the character or monster can walk in 1 round. (PHB)
Also
Speed: A monster’s speed tells you how far it can move on its turn...All creatures have a walking speed, simply called the monster’s speed. Creatures that have no form of ground-based locomotion have a walking speed of 0 feet. (DMG)
So if your character has 30 ft speed, that is a 30 ft walking speed. If they have a 30 ft walking speed they have a 30 foot speed. They are defined in the game to be the same.
When you put the Boots of Speed on, your walking speed doubles. Per the above, that just means that the stat called "speed" on your character sheet gets multiplied by two. In the example above the OP's speed/walking speed started at 45 then becomes 90. This is his new speed.
Jeremy Crawford says:
The rule on swimming doesn't specify that you're using your walking speed. You use whatever speed you can to paddle, kick, and otherwise swim.
This means you can use your fly speed to swim. And your climb speed. And your burrow speed. As long as it is a speed you have, you can use it to swim. Of course, any of these that are not a swim speed will be used at a penalty (each foot of movement costs 1 extra foot), but you can still use them. Obviously this includes walking speed which is the default for most PCs when they go to swim.
Does OP's example character have >=1 speeds? Yes.
What type of speed(s)? A walking speed.
What is their current walking speed? 90 feet.
Can you use walking speed for swimming? Yes.
How far can they swim? 90/2=45 feet.
Simple as that.
In the example above, OP has a walking speed of 45. When he puts on the boots his walking speed becomes 90. THAT IS HIS SPEED. That is how far he can move in a turn. And according to the rules and JC's clarification above you can use that speed to swim.
To those that doubt this, what other number would the character use? His speed is 90 and he has no other types of speed. There is no other number he could use.
So, you would be able to swim 45' (90 feet / 2).
If you walk for 30 feet, you should have 60 feet of walking left as you said. You would then be able to swim for 30 feet (60/2).
If I am playing a race or using an ability that grants me a swimming speed of 30, do the boots affect that?
Not at all.
Jeremy Crawford also said:
Bonuses/penalties to speed apply to your speeds in general, unless the text specifies walking, flying, etc.
Since it specifies walking speed, it only increases your walking speed with no effect on any other speeds you might have.
Example
Just to clarify things let me make an example where your character has a 45ft speed (walking) and 15 ft swim speed.
Putting on boots of speed would mean that you now have a 90ft walking speed and a 15ft swim speed. You could then swim 15ft using just your swim speed. After that you will have 75 feet of movement left since:
If you have more than one speed, such as your walking speed and a flying speed, you can switch back and forth between your speeds during your move. Whenever you switch, subtract the distance you've already moved from the new speed. The result determines how much farther you can move. If the result is 0 or less, you can't use the new speed during the current move. (PHB)
So with the remaining 75 feet of movement, you would be able to go 75/2=37 feet further in the water. This means you would have moved a total of 15+37 = 52 feet in the water.
If the item's description had been written "the boots double your speed", then all of a creature's speeds would be doubled (walking, flying, swimming, etc.). This was clearly not the intent.
So, by writing "double your walking speed" they are specifically indicating that those other types of speeds are not affected by this item. Which makes sense since they are boots and not flippers or jet engines. It does not restrict how and when that walking speed can be used though. If they had intended that, they would have added language to that effect.
You've definitely called out the correct section on movement speeds found in the book, but it's worth noting that in the time since the original printing of the PHB, there was errata added to make this kind of movement more explicitly clear:
Climbing, Swimming, and Crawling
Each foot of movement costs 1 extra foot (2 extra feet in difficult terrain) when you're climbing, swimming, or crawling. You ignore this extra cost if you have a climbing speed and use it to climb, or a swimming speed and use it to swim. At the DM's option, climbing a slippery vertical surface or one with few handholds requires a successful Strength (Athletics) check. Similarly, gaining any distance in rough water might require a successful Strength (Athletics) check.
—Movement, Player's Handbook, pg. 182 (with 2018 Errata)
So what we see here is that the movement cost only applies so long as the creature does not use their movement speed for that particular type of movement. It doesn't say that you're unable to use your walking speed simply because you have a climbing speed.
So if you're using your climbing speed, you'd use 1 foot of movement for each foot you move. Then, if you run out of climbing speed, you can keep going at a rate of 2 feet of movement for each foot you move. As a result, your third option, moving 15 feet with climbing speed, then moving 25 / 2 feet with their remaining walking speed is correct; though it should be noted that Rules-as-Written, there's no requirement to round distances, since 5th Edition does not "snap to a grid" by default. So it's perfectly legal for a character to move exactly 12.5 feet (or in this case, 27.5 feet in total) during their turn.
Best Answer
Charge: http://paizo.com/pathfinderRPG/prd/combat.html#charge
(Edit: Assuming the other requirements are met as the other answer points out, such as any difficult terrain (high winds for example))
"Up to your speed or less"
So long as you have a flight speed, you can charge while flying - similarly for regular moving, climbing, swimming and burrowing. See feats like Death from above for corroboration. http://www.d20pfsrd.com/feats/combat-feats/death-from-above-combat
You can't however charge while making those movement types without the relevant speed.
This is easier to see with climbing or swimming as flying without a flight speed generally gets referred to as 'falling'. But, for example, dropping* downwards from a tree in an ambush wouldn't count as a charge due to lacking the relevant movement speed, though you would probably get +1 for being on higher ground.
(While the climb skill specifically mentions not being able to run while climbing, even with a climb speed, RAW the ability to run or not doesn't actually hinder your ability to charge; RAW the two are different things. Similarly the swim skill limits running to a straight line, even with a swim speed.)
*You could however jump downwards (e.g. Branch Pounce), using the acrobatics skill, though this jumping charge would still need to be in a straight and unobstructed line. With a successful jump check, I would count this as movement made using your regular overland speed.