No.

You cannot take a 5-foot step upwards because doing so would take at least 10 feet of movement, which invalidates the 5-foot-step action and becomes a Move Action instead.

Anytime you cannot spend only 5-foot of your movement to enter a square, it should require a move action, and thus it is no longer a 5-foot step.

You can move 5 feet in any round when you don’t perform any other kind of movement. Taking this 5-foot step never provokes an attack of opportunity. You can’t take more than one 5-foot step in a round, and you can’t take a 5-foot step in the same round that you move any distance.

If you moved 10-foot, it is no longer a 5-foot step. This has been discussed on paizo's messageboards here and here.

According to James Jacobs (Pathfinder's Creative Director), a creature should not be able to fly upwards as a 5-foot step.

The only exception for this, back in 3.5, was creatures that had Perfect maneuvering, which allowed them to move upwards without using double their movement, so they could move 5 feet upwards without using 10 feet of their movement. This exception no longer exists in Pathfinder, but if a creature has a similar ability, the exception should apply.

Yes, movement may continue [at a penalty] when special movement speed has run out

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.