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, it's not stunned ... the way it's worded it tells you what happens on a failed save:
On a failed save, a target takes 8d8 psychic damage and is stunned until the end of your next turn.
and then it tells you what happens on a successful save:
On a successful save, a target takes half as much damage.
Nothing more, nothing less ... no stun mentioned on successful save .. so no stun.
Best Answer
It at most forces one save per turn.
If the creature remains in the greased area, then it might have to save again if it is not already prone. But if it starts its turn in the grease (prone or not) and then exits, it is still difficult terrain. So you want to make sure the enemies will be forced to cross the grease, or at least can't just exit by moving closer to you.
So the only moments a creature must make a save are:
However, you are interpreting the spell wrong.
It is not a spell meant to keep people prone. It is a battlefield control spell. Need to cover the party retreat? Grease the tunnel behind the last person. Want to make the enemies take longer to get to melee range? Grease between you and them.
Enemies will lose 20ft of movement to cross the greased area in the best scenario (for the enemies).
Also notice it does not use up concentration, and this is a big plus.
There is this video by nerdarchy about using the spell.Except they got it wrong in the first segment. Grease is not flammable. And they got the burning hands spell damage wrong.