Let's say you're a 40 speed, 15 climbing speed creature. There's an initial ledge at 20 feet, but you're trying to reach the top at 25 feet.

From the Player's Handbook, page 182:

While climbing or swimming, each foot of movement costs 1 extra foot (2 extra feet in difficult terrain), unless a creature has a climbing or swimming speed. At the GM’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.

However, on page 190:

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.

Combined with this information, there may be an odd scenario where a character has moved their maximum climbing speed.

Can they:

1. Move the 15 feet up the cliff, and cannot reach neither the ledge or the top?


2. Move their normal speed at twice the cost, allowing them to climb up to 20 feet in this turn and reach the ledge? (40/2= 20 feet)


3. Move the 15 feet of climbing speed up the cliff, and are able to spend the rest of their speed (40-15 = 25 left) at the extra foot cost, since they "no longer have a climbing speed", putting them at the very top of the cliff (25/2=10 [rounded down]), 15+10 =25 feet)?

You can keep climbing if you have walking movement left

You have quoted the relevant text, which allows you to "switch back and forth between your speeds during your move". Therefore, the third option you present is accurate. You first move up to your climbing speed, then spend two feet for every one foot of your walking speed to keep climbing, leaving you at the top of the cliff with no remaining movement.

A strict RAW reading of "unless a creature has a climbing or swimming speed" would say that a creature with a climbing speed never has to spend extra movement to climb. A creature with a climbing speed less than their walking speed, like a Tabaxi PC, would be nonsensical - their walking speed could be used to climb without penalty, so why list a lower climbing speed? The only reasonable interpretation is that "unless a creature has a climbing or swimming speed" is intended to only apply while that climbing or swimming speed is being used; therefore, a PC can keep climbing with a penalty if they have walking movement left.

You may need to start making Athletics checks

If your DM allowed you to skip an Athletics check due to having a climbing speed, it may be reasonable for them to change their mind if you switch to using your walking speed. Mike Mearls says that you won't need an Athletics check if you have a climbing speed in the following Sage Advice tweet:

@JeremyECrawford @mikemearls Do creatures with a climb speed make Athletics checks? If so, Advantage would be nice-lets make that official.

— DnD Sphinx (@DnDSphinx)

no check needed to climb

— Mike Mearls (@mikemearls)

That said, even creatures with a climbing speed may need to make an Athletics check to climb "a slippery vertical surface or one with few handholds" (PHB, p. 182). Jeremy Crawford indirectly confirms this in the following Sage Advice tweet:

@JeremyECrawford Is it RAI that creatures with a climb speed need to make Athletics checks to climb? RAW are silent on the issue, thus yes?

— Shawn Dry (@ProfSiccus)

A trait like Spider Climb, not a climbing speed, lets some creatures forgo ability checks when climbing.

— Jeremy Crawford (@JeremyECrawford)

It should be Permissible

You've definitely called out the correct section on movement speeds found in the book:

Climbing, Swimming, and Crawling

While climbing or swimming, each foot of movement costs 1 extra foot (2 extra feet in difficult terrain), unless a creature has a climbing or swimming speed. 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.

So what we see here is that the movement cost only applies so long as the creature does not have a movement speed for that particular type of movement. It doesn't infer that you're unable to use your walking speed simply because you have a climbing speed.

There is an ambiguity that needs to be resolved though. The wording says "unless a creature has a climbing speed". It doesn't specifically say that creature needs to have unused climbing speed. One possible reading is that because this character has a climbing speed, they never suffer movement penalties when climbing, even if climbing will exceed their listed climbing speed. So in this scenario, they'd be able to climb up to 40', because their normal movement speed wouldn't be penalized.

I suspect that what is intended is the way you've described it; where at the point they've already climbed 15', they are intended to be treated as no longer having a climbing speed [for the rest of their turn]. As a result, they would then have to spend twice their movement speed per foot of movement, and only travel an additional 12.5' while climbing—note that 5e does not default to having a grid, and you wouldn't round character movement; it's fully possible to move exactly 12.5 feet, or 27.5' in total when combined with your first 15' movement.

One additional reason to think that this is the intended way to rule this is that in the Monster Manual, there are many creatures (especially bears) that are specified with both a regular movement speed and a climbing speed, and with climbing speeds that are less than their regular movement speeds. If the RAW ruling I interpreted above were intended, then there would be no valid reason to print creatures with movement speeds like this, since their 40' total movement speed would override their 30' limited climbing speed. So the method you described in option 3 is probably the intended way to make this ruling.

But because of this ambiguity in RAW, the ruling will have to come down to your DM.

No, when you run out of climb speed, you're done moving.

In the introduction to the Monster Manual (page 8), it has this to say about climb speed:

CLIMB

A monster that has a climbing speed can use all or part of its movement to move on vertical surfaces. The monster doesn't need to spend extra movement to climb.

And you quoted the other relevant text (emphasis mine):

While climbing or swimming, each foot of movement costs 1 extra foot, unless a creature has a climbing or swimming speed.

Based on those, my interpretation is that having a climb speed completely overrides the normal climb rules and replaces them; you climb at the listed speed instead of being able to spend 2 feet of movement per foot of climbing -- in other words, the climb speed replaces the '1 extra foot' rule and you no longer have the option of climbing by using additional base speed. You must use that climb speed instead of your walking speed when climbing, and once that's used up, you're done climbing. You can't convert your normal speed into additional climbing, even if it would be better.

But it won't be better. In every case I can find, a creature's climb speed is more than half its walking speed, so using their climb speed is the strictly superior choice. In most cases, the climb speed is equal to move speed, but in the few where it isn't (such as the Panther), the climb speed is still greater than half their movement.

So, for example, if the aforementioned panther climbed a 40-foot cliff, it could then switch to normal movement, and move an additional 10 feet (by the 'switch back and forth' rule), but if the cliff is 45 feet tall, the panther is going to have to just hang out there until next turn (unless it took a Dash action, which essentially doubles all your speeds).

If there is a scenario where this isn't the case, I might change my mind, but I haven't been able to find one.