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}\$.
If the fighter carries the warlock
He is carrying:
- 56 lb of his own gear
- 242 lb of half-orc
- 10 lb of warlock gear
Total: 308 lb versus a carrying capacity of 255 lb so he's going nowhere.
If the fighter pushes, drags or lifts the warlock
He is still carrying 56 lb of his own gear which is less than the 85 lb for being encumbered so he isn't.
He can easily push, drag or lift the warlock because 245 lb is less than 255 lb. If he pushes or drags him, his speed is unaffected because what he is pushing or dragging is less than his carrying capacity. If he lifts him, he can't move because he is now exceeding his maximum carrying capacity.
TL;DR
Whatever you are carrying goes to your encumbrance - this affects your speed. If you are carrying stuff and also someone who is carrying more stuff then you are carrying all of that.
In addition, you can push or drag up to your carrying capacity without changing your (encumbered) speed, however, if you push or drag more than this (up to twice as much) your speed drops to 5 feet.
At the absolute maximum, you can carry 15 times your strength score and push/drag twice that much again.
Best Answer
The creature can move at half speed to avoid both effects.
Neither ball bearings nor caltrops state that a creature's speed is physically penalized, just that a creature can choose to move at half speed to avoid their effects. Then we can handle each consideration separately.
First, ball bearings (emphasis mine):
Is the creature moving through the area of ball bearings at half speed? Then they are unaffected.
Next, caltrops (emphasis mine):
Is the creature moving through the area of caltrops at half speed? Then they are unaffected.
An area covered in both ball bearings and caltrops is no different. Is the creature moving through the area at half speed? Then they are unaffected.
Note that if the creature chose to move greater than half their speed across the combined area then they would be forced to make both saving throws, one against the ball bearings and one against the caltrops, possibly suffering one effect or the other effect or both effects combined.