There is a rule in the "Moving Around Other Creatures" section which states:

Whether a creature is a friend or an enemy, you can't willingly end your move in its space...

I am wondering just how absolute this rule is and have come up with example situations where its application is unclear to me:

1. On your turn, you are forced to occupy the same space as another creature, you have no more movement available, but you still have your action or bonus action available. Are you required to take the Dash action or to use some method (such as casting misty step) to get you out of the creature's space?



2. 
   

   Caltrops state:

   
   
   

   
   

   Any creature that enters the area must succeed on a DC 15 dexterity saving throw or stop moving and take 1 piercing damage. Until the creature regains at least 1 hit point, its walking speed is reduced by 10 feet. A creature moving through the area at half speed doesn't need to make the saving throw.

   
   

   
   
   

   An adjacent creature in standing on top of caltrops, and you only have 15 speed (and 15 movement left). I believe that you could not walk through them at half speed as you would run out of movement and be stuck in the creature's space.
   
   However, could you walk through them normally even though they have a chance of having your move stop in the creature's space (which, presumably, the character would know about, and thus would make it a willing decision)?

   
   



3. Everything is difficult terrain, there is an adjacent creature, and you have 15 feet of movement left, could you move onto the creature's space and then realize you can't leave that space (as you don't have enough movement to do so)?

I believe that my question is different from this one in a few ways. My second and third examples involve more willingly ending your move in another creature's space (at least to me they seem to be more willingly done) and my first example calls into question what happens when the creature still has a way to gain the ability to leave the creature's space (such as by taking the Dash action or casting misty step) which the linked question did not mention or address.

Does the rule preventing you from ending your turn in another creature's space force you to take certain actions?

The rule requires that you not willingly end your turn in another creature's space. Sometimes you will be prevented from taking actions in order to make sure the rule holds true.

The DM always has jurisdiction over edge-cases like this and will often make rulings on consequences and benefits, so although I speak from the perspective of being a DM myself, my own interpretations will not always hold true.

Willing and unwilling do not have mechanical definitions, but here's how I would rule them for practical use.

Unwilling:

• Effects that have a risk but not a guarantee of forcing your turn to end in that space

Willing:

• Effects that guarantee you will end your turn in that space which are not hidden from player knowledge


• Effects coordinated with a party member to place you in a space.

Regarding your examples:

1. 
   

   You must not willingly spend movement to end in the creature's space. In the case that they're forced unwillingly into the space after/before movement, such as by a grapple, RAW does not require action economy (misty step, breaking grapples, dashes, etc) to be spent on vacating the space.

   
   
   

   
   

   Sidenote to 1: The rules are unclear whether a player may receive their once-per-turn movement and not spend it to remain occupying the space, but this is clearly against the spirit of the rule. If a player has movement, I believe it should be spent to vacate the space. I also believe that actions a player controls (such as misty step and falling from above) should not cause the space to be occupied, but I would rule these on a case-by-case basis.

   
   

   
   



2. This is a tougher one to rule on. RAW you can attempt to spend movement to pass through the space and unwillingly have your movement reduced to 0 in that space. The creature must be friendly or your size class has to permit movement through it's space (size class overrules this general rule).




3. This would likely require a rollback. You willingly spent movement and then realized it was movement you couldn't use. You broke a rule, the DM will likely undo the action that broke it or offer you an alternative (such as taking disadvantage).

No, the rule can't force you to take certain actions

1. You don't have to take the Dash action. The rule only states you can't willingly end your movement in another creature's space. It says nothing about having to get out of such a situation if forced into it. Be warned that a DM might choose to apply consequences such as disadvantage on attacks if you are jostling with another creature for room. Keep in mind also it says movement, not turn. You can start and end your turn in another creature's space and only be required to leave it if you move and can get clear.




2. Moving through the same tile as another creature is considered to be moving through difficult terrain anyway, which means it costs you twice as much movement. It would cost all of your 10 feet of movement just to enter that tile which isn't allowed. If you had 20 feet of movement, you could try moving through their tile and risk getting stuck because you aren't willingly stopping.




3. Only if you aren't aware that it is difficult terrain, e.g. the Spike Growth spell.

When you enter the allied creature's space, you have to have an exit strategy. Something might happen during your turn that negates that strategy, such as caltrops, the Sentinel feat, or being knocked prone, but you can't go in without having a way out.

That said, if your exit strategy gets screwed up or you get forced into somebody's space with no movement left, you're not actually required to do anything to fix it.

So in your case #1, you were somehow forced to occupy the same space during your turn with no movement left (perhaps an enemy creature's reaction had a push effect). You are not required to spend an action to dash. If you had enough movement left to move to any other square, you would be compelled to do so, but nothing requires you to take an action (or do anything, for that matter) to get yourself out of that square (but the DM might decide that being jammed up against your ally brings penalties -- more on that later).

In case #2, you can enter the square with the intent of leaving, and if you fail your save against the caltrops, you're stuck there. The same goes for entering the area of a number of spells such as web or Evard's black tentacles.

In case #3, the only way you can enter the space is if you are somehow unaware of the difficult terrain effect that would prevent you from leaving with your remaining movement, perhaps due to hallucinatory terrain or some such effect. If you don't have a clear way out, you can't go in. By the way, do remember the often-ignored rule from the Difficult Terrain section in the Player's Handbook, page 190:

The space of another creature, whether hostile or not, also counts as difficult terrain.

Your DM might decide to apply a penalty of some sort to two creatures sharing the same space. It's not, strictly speaking, in the rules, but the DM could certainly decide that you both take disadvantage on your attacks and attackers have advantage against you as you get in each other's way.

Willing movement means movement you take on your turn as part of your set of activities, as opposed to movement forced upon the character as a result of someone else's activities.

For example, if a spell effect were to push/pull you 10 feet, the prohibition on ending your movement in another creature's space would not apply, because it is forced movement, not willing movement.

If your own movement will cause you to end your movement where you cannot, then you either cannot take that movement, or you must extend your movement in some manner to make the movement legal.