PH Page 190 has a section on different speeds that I think would apply:
If you have more than one 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 ar less, you can't use the new speed during the current move.
Your Questions
If a creature with normal speed 30 entered the area after moving 15 feet, what is its remaining available movement?
You have a speed of 30 and move 15 feet. You then enter the area and you now have a move of 15 feet. Your already-traveled movement is subtracted from your move speed, which is 0 or less, so you can't use the "new" speed.
If a creature with normal speed 30/halved speed 15 starts in the area and moves 15 feet to exit the area, what is its remaining movement?
This is the opposite, you have a speed of 15, and move 15 feet and are out of the area. Your speed is now 30, and you've only used 15 feet, so have 15 more feet available.
Difficult Terrain
The section you are referring to is under the Speed heading, which states that "[t]he following rules determine how far a character or monster can move in a minute, an hour, or a day." Therefore, those rules wouldn't apply to in-combat movement since that has its own rules.
The rules for in-combat difficult terrain are found PH page 190:
Every foot of movement in difficult terrain costs 1 extra foot. This rule is true even if multiple things in a space count as difficult terrain.
Nothing about halved movement. Hence, I conclude that halved movement, at least in combat (since the spell in question has a duration of 10min, I don't see it being that useful outside of combat), means \$ \frac{speed}{2}\$.
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.
Best Answer
The speed reduction does not stack.
The rules for combining game effects state (DMG p. 252, added in the DMG errata; emphasis mine):
So you cannot be affected by the speed reduction of more than one set of caltrops at a time.
But mundane items aren't on the list!
Indeed, "mundane items" is not on the list of examples given in the rule quoted above. This list is not meant to be exhaustive. To demonstrate this, we can observe that the official rules answers document, the Sage Advice Compendium, tells us that an opportunity attack is a game feature:
Opportunity attacks, a type of reaction, are not "spells, class features, feats, racial traits, monster abilities, and magic items", so the list of game features given in the "same name" rule is not exhaustive.
However, the application of the rule needs to be exhaustive: excluding mundane items from this rule breaks the game. Alchemist's Fire is a mundane item:
If we allow this feature to stack with itself, any creature can be quickly overwhelmed by stacking Alchemist's Fire effects. Let's walk through this:
The damage just piles on, and the BBEG cannot keep up with the effects being added on.